Permutation groups#

A permutation group is a finite group \(G\) whose elements are permutations of a given finite set \(X\) (i.e., bijections \(X \longrightarrow X\)) and whose group operation is the composition of permutations. The number of elements of \(X\) is called the degree of \(G\).

In Sage, a permutation is represented as either a string that defines a permutation using disjoint cycle notation, or a list of tuples, which represent disjoint cycles. That is:

(a,...,b)(c,...,d)...(e,...,f)  <--> [(a,...,b), (c,...,d),..., (e,...,f)]
                  () = identity <--> []

You can make the “named” permutation groups (see permgp_named.py) and use the following constructions:

permutation group generated by elements,
direct_product_permgroups, which takes a list of permutation groups and returns their direct product.

JOKE: Q: What’s hot, chunky, and acts on a polygon? A: Dihedral soup. Renteln, P. and Dundes, A. “Foolproof: A Sampling of Mathematical Folk Humor.” Notices Amer. Math. Soc. 52, 24-34, 2005.

Index of methods#

Here are the methods of a PermutationGroup()

`as_finitely_presented_group()`	Return a finitely presented group isomorphic to `self`.
`blocks_all()`	Return the list of block systems of imprimitivity.
`cardinality()`	Return the number of elements of this group.
`center()`	Return the subgroup of elements that commute with every element of this group.
`centralizer()`	Return the centralizer of `g` in `self`.
`character()`	Return a group character from `values`, where `values` is a list of the values of the character evaluated on the conjugacy classes.
`character_table()`	Return the matrix of values of the irreducible characters of a permutation group \(G\) at the conjugacy classes of \(G\).
`cohomology()`	Computes the group cohomology \(H^n(G, F)\), where \(F = \ZZ\) if \(p=0\) and \(F = \ZZ / p \ZZ\) if \(p > 0\) is a prime.
`cohomology_part()`	Compute the p-part of the group cohomology \(H^n(G, F)\), where \(F = \ZZ\) if \(p=0\) and \(F = \ZZ / p \ZZ\) if \(p > 0\) is a prime.
`commutator()`	Return the commutator subgroup of a group, or of a pair of groups.
`composition_series()`	Return the composition series of this group as a list of permutation groups.
`conjugacy_class()`	Return the conjugacy class of `g` inside the group `self`.
`conjugacy_classes()`	Return a list with all the conjugacy classes of `self`.
`conjugacy_classes_representatives()`	Return a complete list of representatives of conjugacy classes in a permutation group \(G\).
`conjugacy_classes_subgroups()`	Return a complete list of representatives of conjugacy classes of subgroups in a permutation group \(G\).
`conjugate()`	Return the group formed by conjugating `self` with `g`.
`construction()`	Return the construction of `self`.
`cosets()`	Return a list of the cosets of `S` in `self`.
`degree()`	Return the degree of this permutation group.
`derived_series()`	Return the derived series of this group as a list of permutation groups.
`direct_product()`	Wraps GAP’s `DirectProduct`, `Embedding`, and `Projection`.
`domain()`	Return the underlying set that this permutation group acts on.
`exponent()`	Computes the exponent of the group.
`fitting_subgroup()`	Return the Fitting subgroup of `self`.
`fixed_points()`	Return the list of points fixed by `self`, i.e., the subset of `.domain()` not moved by any element of `self`.
`frattini_subgroup()`	Return the Frattini subgroup of `self`.
`gen()`	Return the \(i\)-th generator of `self`; that is, the \(i\)-th element of the list `self.gens()`.
`gens()`	Return tuple of generators of this group.
`gens_small()`	For this group, returns a generating set which has few elements.
`group_id()`	Return the ID code of this group, which is a list of two integers.
`group_primitive_id()`	Return the index of this group in the GAP database of primitive groups.
`has_element()`	Return whether `item` is an element of this group - however ignores parentage.
`holomorph()`	The holomorph of a group as a permutation group.
`homology()`	Computes the group homology \(H_n(G, F)\), where \(F = \ZZ\) if \(p=0\) and \(F = \ZZ / p \ZZ\) if \(p > 0\) is a prime. Wraps HAP’s `GroupHomology` function, written by Graham Ellis.
`homology_part()`	Computes the \(p\)-part of the group homology \(H_n(G, F)\), where \(F = \ZZ\) if \(p=0\) and \(F = \ZZ / p \ZZ\) if \(p > 0\) is a prime. Wraps HAP’s `Homology` function, written by Graham Ellis, applied to the \(p\)-Sylow subgroup of \(G\).
`id()`	Return the ID code of this group, which is a list of two integers.
`intersection()`	Return the permutation group that is the intersection of `self` and `other`.
`irreducible_characters()`	Return a list of the irreducible characters of `self`.
`is_cyclic()`	Return `True` if this group is cyclic.
`is_elementary_abelian()`	Return `True` if this group is elementary abelian. An elementary abelian group is a finite abelian group, where every nontrivial element has order \(p\), where \(p\) is a prime.
`is_isomorphic()`	Return `True` if the groups are isomorphic.
`is_monomial()`	Return `True` if the group is monomial. A finite group is monomial if every irreducible complex character is induced from a linear character of a subgroup.
`is_nilpotent()`	Return `True` if this group is nilpotent.
`is_normal()`	Return `True` if this group is a normal subgroup of `other`.
`is_perfect()`	Return `True` if this group is perfect. A group is perfect if it equals its derived subgroup.
`is_pgroup()`	Return `True` if this group is a \(p\)-group.
`is_polycyclic()`	Return `True` if this group is polycyclic. A group is polycyclic if it has a subnormal series with cyclic factors. (For finite groups, this is the same as if the group is solvable - see `is_solvable()`.)
`is_primitive()`	Return `True` if `self` acts primitively on `domain`.
`is_regular()`	Return `True` if `self` acts regularly on `domain`.
`is_semi_regular()`	Return `True` if `self` acts semi-regularly on `domain`.
`is_simple()`	Return `True` if the group is simple.
`is_solvable()`	Return `True` if the group is solvable.
`is_subgroup()`	Return `True` if `self` is a subgroup of `other`.
`is_supersolvable()`	Return `True` if the group is supersolvable.
`is_transitive()`	Return `True` if `self` acts transitively on `domain`.
`isomorphism_to()`	Return an isomorphism from `self` to `right` if the groups are isomorphic, otherwise `None`.
`isomorphism_type_info_simple_group()`	If the group is simple, then this returns the name of the group.
`iteration()`	Return an iterator over the elements of this group.
`largest_moved_point()`	Return the largest point moved by a permutation in this group.
`list()`	Return list of all elements of this group.
`lower_central_series()`	Return the lower central series of this group as a list of permutation groups.
`maximal_normal_subgroups()`	Return the maximal proper normal subgroups of `self`.
`minimal_generating_set()`	Return a minimal generating set
`minimal_normal_subgroups()`	Return the nontrivial minimal normal subgroups `self`.
`molien_series()`	Return the Molien series of a permutation group. The function
`ngens()`	Return the number of generators of `self`.
`non_fixed_points()`	Return the list of points not fixed by `self`, i.e., the subset of `self.domain()` moved by some element of `self`.
`normal_subgroups()`	Return the normal subgroups of this group as a (sorted in increasing order) list of permutation groups.
`normalizer()`	Return the normalizer of `g` in `self`.
`normalizes()`	Return `True` if the group `other` is normalized by `self`.
`poincare_series()`	Return the Poincaré series of \(G \mod p\) (\(p \geq 2\) must be a prime), for \(n\) large.
`random_element()`	Return a random element of this group.
`representative_action()`	Return an element of self that maps \(x\) to \(y\) if it exists.
`semidirect_product()`	The semidirect product of `self` with `N`.
`sign_representation()`	Return the sign representation of `self` over `base_ring`.
`socle()`	Return the socle of `self`.
`solvable_radical()`	Return the solvable radical of `self`.
`stabilizer()`	Return the subgroup of `self` which stabilize the given position. `self` and its stabilizers must have same degree.
`strong_generating_system()`	Return a Strong Generating System of `self` according the given base for the right action of `self` on itself.
`structure_description()`	Return a string that tries to describe the structure of `G`.
`subgroup()`	Wraps the `PermutationGroup_subgroup` constructor. The argument `gens` is a list of elements of `self`.
`subgroups()`	Return a list of all the subgroups of `self`.
`sylow_subgroup()`	Return a Sylow \(p\)-subgroup of the finite group \(G\), where \(p\) is a prime.
`transversals()`	If G is a permutation group acting on the set \(X = \{1, 2, ...., n\}\) and H is the stabilizer subgroup of <integer>, a right (respectively left) transversal is a set containing exactly one element from each right (respectively left) coset of H. This method returns a right transversal of `self` by the stabilizer of `self` on <integer> position.
`trivial_character()`	Return the trivial character of `self`.
`upper_central_series()`	Return the upper central series of this group as a list of permutation groups.

AUTHORS:

David Joyner (2005-10-14): first version
David Joyner (2005-11-17)
William Stein (2005-11-26): rewrite to better wrap Gap
David Joyner (2005-12-21)
William Stein and David Joyner (2006-01-04): added conjugacy_class_representatives
David Joyner (2006-03): reorganization into subdirectory perm_gps; added __contains__, has_element; fixed _cmp_; added subgroup class+methods, PGL,PSL,PSp, PSU classes,
David Joyner (2006-06): added PGU, functionality to SymmetricGroup, AlternatingGroup, direct_product_permgroups
David Joyner (2006-08): added degree, ramification_module_decomposition_modular_curve and ramification_module_decomposition_hurwitz_curve methods to PSL(2,q), MathieuGroup, is_isomorphic
Bobby Moretti (2006)-10): Added KleinFourGroup, fixed bug in DihedralGroup
David Joyner (2006-10): added is_subgroup (fixing a bug found by Kiran Kedlaya), is_solvable, normalizer, is_normal_subgroup, Suzuki
David Kohel (2007-02): fixed __contains__ to not enumerate group elements, following the convention for __call__
David Harvey, Mike Hansen, Nick Alexander, William Stein (2007-02,03,04,05): Various patches
Nathan Dunfield (2007-05): added orbits
David Joyner (2007-06): added subgroup method (suggested by David Kohel), composition_series, lower_central_series, upper_central_series, cayley_table, quotient_group, sylow_subgroup, is_cyclic, homology, homology_part, cohomology, cohomology_part, poincare_series, molien_series, is_simple, is_monomial, is_supersolvable, is_nilpotent, is_perfect, is_polycyclic, is_elementary_abelian, is_pgroup, gens_small, isomorphism_type_info_simple_group. moved all the”named” groups to a new file.
Nick Alexander (2007-07): move is_isomorphic to isomorphism_to, add from_gap_list
William Stein (2007-07): put is_isomorphic back (and make it better)
David Joyner (2007-08): fixed bugs in composition_series, upper/lower_central_series, derived_series,
David Joyner (2008-06): modified is_normal (reported by W. J. Palenstijn), and added normalizes
David Joyner (2008-08): Added example to docstring of cohomology.
Simon King (2009-04): __cmp__ methods for PermutationGroup_generic and PermutationGroup_subgroup
Nicolas Borie (2009): Added orbit, transversals, stabiliser and strong_generating_system methods
Christopher Swenson (2012): Added a special case to compute the order efficiently. (This patch Copyright 2012 Google Inc. All Rights Reserved. )
Javier Lopez Pena (2013): Added conjugacy classes.
Sebastian Oehms (2018): added _coerce_map_from_ in order to use isomorphism coming up with as_permutation_group method (Issue #25706)
Christian Stump (2018): Added alternative implementation of strong_generating_system directly using GAP.
Sebastian Oehms (2018): Added PermutationGroup_generic._Hom_() to use sage.groups.libgap_morphism.GroupHomset_libgap and PermutationGroup_generic.gap() and PermutationGroup_generic._subgroup_constructor() (for compatibility to libgap framework, see github issue #26750

REFERENCES:

Cameron, P., Permutation Groups. New York: Cambridge University Press, 1999.
Wielandt, H., Finite Permutation Groups. New York: Academic Press, 1964.
Dixon, J. and Mortimer, B., Permutation Groups, Springer-Verlag, Berlin/New York, 1996.

Note

Though Suzuki groups are okay, Ree groups should not be wrapped as permutation groups - the construction is too slow - unless (for small values or the parameter) they are made using explicit generators.

sage.groups.perm_gps.permgroup.PermutationGroup(gens=None, *args, **kwds)#

Return the permutation group associated to \(x\) (typically a list of generators).

INPUT:

gens – (default: None) list of generators
gap_group – (optional) a gap permutation group
canonicalize – boolean (default: True); if True, sort generators and remove duplicates

OUTPUT:

a permutation group

EXAMPLES:

sage: G = PermutationGroup([[(1,2,3),(4,5)],[(3,4)]])
sage: G
Permutation Group with generators [(3,4), (1,2,3)(4,5)]

We can also make permutation groups from PARI groups:

sage: # needs sage.libs.pari
sage: H = pari('x^4 - 2*x^3 - 2*x + 1').polgalois()
sage: G = PariGroup(H, 4); G
PARI group [8, -1, 3, "D(4)"] of degree 4
sage: H = PermutationGroup(G); H
Transitive group number 3 of degree 4
sage: H.gens()
((1,2,3,4), (1,3))

We can also create permutation groups whose generators are GAP permutation objects:

sage: p = gap('(1,2)(3,7)(4,6)(5,8)'); p
(1,2)(3,7)(4,6)(5,8)
sage: PermutationGroup([p])
Permutation Group with generators [(1,2)(3,7)(4,6)(5,8)]

Permutation groups can work on any domain. In the following examples, the permutations are specified in list notation, according to the order of the elements of the domain:

sage: list(PermutationGroup([['b','c','a']], domain=['a','b','c']))
[(), ('a','b','c'), ('a','c','b')]
sage: list(PermutationGroup([['b','c','a']], domain=['b','c','a']))
[()]
sage: list(PermutationGroup([['b','c','a']], domain=['a','c','b']))
[(), ('a','b')]

There is an underlying gap object that implements each permutation group:

sage: G = PermutationGroup([[(1,2,3,4)]])
sage: G._gap_()
Group( [ (1,2,3,4) ] )
sage: gap(G)
Group( [ (1,2,3,4) ] )
sage: gap(G) is G._gap_()
True
sage: G = PermutationGroup([[(1,2,3),(4,5)],[(3,4)]])
sage: current_randstate().set_seed_gap()
sage: G1, G2 = G._gap_().DerivedSeries()
sage: G1
Group( [ (3,4), (1,2,3)(4,5) ] )
sage: G2.GeneratorsSmallest()
[ (3,4,5), (2,3)(4,5), (1,2)(4,5) ]

We can create a permutation group from a group action:

sage: a = lambda x: (2*x) % 7
sage: H = PermutationGroup(action=a, domain=range(7))                           # needs sage.combinat
sage: H.orbits()                                                                # needs sage.libs.pari
((0,), (1, 2, 4), (3, 6, 5))
sage: H.gens()                                                                  # needs sage.libs.pari
((1,2,4), (3,6,5))

Note that we provide generators for the acting group. The permutation group we construct is its homomorphic image:

sage: # needs sage.combinat
sage: a = lambda g, x: vector(g*x, immutable=True)
sage: X = [vector(x, immutable=True) for x in GF(3)^2]
sage: G = SL(2,3); G.gens()
(
[1 1]  [0 1]
[0 1], [2 0]
)
sage: H = PermutationGroup(G.gens(), action=a, domain=X)
sage: H.orbits()
(((0, 0),), ((1, 0), (2, 0), (0, 1), (1, 1), (2, 1), (0, 2), (1, 2), (2, 2)))
sage: H.gens()
(((0,1),(1,1),(2,1))((0,2),(2,2),(1,2)),
 ((1,0),(0,2),(2,0),(0,1))((1,1),(1,2),(2,2),(2,1)))

The orbits of the conjugation action are the conjugacy classes, i.e., in bijection with integer partitions:

sage: a = lambda g, x: g*x*g^-1
sage: [len(PermutationGroup(SymmetricGroup(n).gens(), action=a,                 # needs sage.combinat
....:                       domain=SymmetricGroup(n)).orbits())
....:  for n in range(1, 8)]
[1, 2, 3, 5, 7, 11, 15]

class sage.groups.perm_gps.permgroup.PermutationGroup_action(gens, action, domain, gap_group=None, category=None, canonicalize=None)#

Bases: PermutationGroup_generic

A permutation group given by a finite group action.

EXAMPLES:

A cyclic action:

sage: n = 3
sage: a = lambda x: SetPartition([[e % n + 1 for e in b] for b in x])
sage: S = SetPartitions(n)                                                      # needs sage.combinat
sage: G = PermutationGroup(action=a, domain=S)                                  # needs sage.combinat
sage: G.orbits()                                                                # needs sage.combinat
(({{1}, {2}, {3}},),
 ({{1, 2}, {3}}, {{1}, {2, 3}}, {{1, 3}, {2}}),
 ({{1, 2, 3}},))

The regular action of the symmetric group:

sage: a = lambda g, x: g*x*g^-1
sage: S = SymmetricGroup(3)
sage: G = PermutationGroup(S.gens(), action=a, domain=S)                        # needs sage.combinat
sage: G.orbits()                                                                # needs sage.combinat
(((),), ((1,3,2), (1,2,3)), ((2,3), (1,3), (1,2)))

The trivial action of the symmetric group:

sage: PermutationGroup(SymmetricGroup(3).gens(),                                # needs sage.combinat
....:                  action=lambda g, x: x, domain=[1])
Permutation Group with generators [()]

orbits()#

Returns the orbits of the elements of the domain under the default group action.

EXAMPLES:

sage: a = lambda x: (2*x) % 7
sage: G = PermutationGroup(action=a, domain=range(7))                       # needs sage.combinat
sage: G.orbits()                                                            # needs sage.combinat
((0,), (1, 2, 4), (3, 6, 5))

class sage.groups.perm_gps.permgroup.PermutationGroup_generic(gens=None, gap_group=None, canonicalize=True, domain=None, category=None)#

Bases: FiniteGroup

A generic permutation group.

EXAMPLES:

sage: G = PermutationGroup([[(1,2,3),(4,5)],[(3,4)]])
sage: G
Permutation Group with generators [(3,4), (1,2,3)(4,5)]
sage: G.center()
Subgroup generated by [()] of
 (Permutation Group with generators [(3,4), (1,2,3)(4,5)])
sage: G.group_id()
[120, 34]
sage: n = G.order(); n
120
sage: G = PermutationGroup([[(1,2,3),(4,5)],[(3,4)]])
sage: TestSuite(G).run()

Element#: alias of PermutationGroupElement

Subgroup#: alias of PermutationGroup_subgroup

as_finitely_presented_group(reduced=False)#

Return a finitely presented group isomorphic to self.

This method acts as wrapper for the GAP function IsomorphismFpGroupByGenerators, which yields an isomorphism from a given group to a finitely presented group.

INPUT:

reduced – (default False) if True, FinitelyPresentedGroup.simplified is called, attempting to simplify the presentation of the finitely presented group to be returned.

OUTPUT:

Finite presentation of self, obtained by taking the image of the isomorphism returned by the GAP function IsomorphismFpGroupByGenerators.

ALGORITHM:

Uses GAP.

EXAMPLES:

sage: CyclicPermutationGroup(50).as_finitely_presented_group()
Finitely presented group < a | a^50 >
sage: DihedralGroup(4).as_finitely_presented_group()
Finitely presented group < a, b | b^2, a^4, (b*a)^2 >
sage: GeneralDihedralGroup([2,2]).as_finitely_presented_group()
Finitely presented group < a, b, c | a^2, b^2, c^2, (c*b)^2, (c*a)^2, (b*a)^2 >

GAP algorithm is not guaranteed to produce minimal or canonical presentation:

sage: G = PermutationGroup(['(1,2,3,4,5)', '(1,5)(2,4)'])
sage: G.is_isomorphic(DihedralGroup(5))
True
sage: K = G.as_finitely_presented_group(); K
Finitely presented group < a, b | b^2, (b*a)^2, b*a^-3*b*a^2 >
sage: K.as_permutation_group().is_isomorphic(DihedralGroup(5))
True

We can attempt to reduce the output presentation:

sage: H = PermutationGroup(['(1,2,3,4,5)', '(1,3,5,2,4)'])
sage: H.as_finitely_presented_group()
Finitely presented group < a, b | b^-2*a^-1, b*a^-2 >
sage: H.as_finitely_presented_group(reduced=True)
Finitely presented group < a | a^5 >

AUTHORS:

Davis Shurbert (2013-06-21): initial version

base(seed=None)#

Return a (minimum) base of this permutation group.

A base \(B\) of a permutation group is a subset of the domain of the group such that the only group element stabilizing all of \(B\) is the identity.

INPUT:

seed (optional, default: None), if given must be a subset of the domain of a base. When used, an attempt to create a base containing all or part of seed will be made.

EXAMPLES:

sage: G = PermutationGroup([(1,2,3),(6,7,8)])
sage: G.base()
[1, 6]
sage: G.base([2])
[2, 6]

sage: H = PermutationGroup([('a','b','c'),('a','y')])
sage: H.base()
['a', 'b', 'c']

sage: S = SymmetricGroup(13)
sage: S.base()
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]

sage: S = MathieuGroup(12)
sage: S.base()
[1, 2, 3, 4, 5]
sage: S.base([1,3,5,7,9,11]) # create a base for M12 with only odd integers
[1, 3, 5, 7, 9]

blocks_all(representatives=True)#

Return the list of block systems of imprimitivity.

For more information on primitivity, see the Wikipedia article on primitive group actions.

INPUT:

representative – (boolean) whether to return all possible block systems of imprimitivity or only one of their representatives (the block can be obtained from its representative set \(S\) by computing the orbit of \(S\) under self).

This parameter is set to True by default (as it is GAP’s default behaviour).

OUTPUT:

This method returns a description of all block systems. Hence, the output is a “list of lists of lists” or a “list of lists” depending on the value of representatives. A bit more clearly, output is:

A list of length (#number of different block systems) of
- block systems, each of them being defined as
  - If representatives=True: a list of representatives of each set of the block system
  - If representatives=False: a partition of the elements defining an imprimitivity block.

See also: degree().

EXAMPLES:

sage: G = PermutationGroup([[(1,2,3),(4,5)], [(1,2)]])
sage: G.order()
12
sage: G = PermutationGroup([()])
sage: G.order()
1
sage: G = PermutationGroup([])
sage: G.order()
1

cardinality() is just an alias:

sage: PermutationGroup([(1,2,3)]).cardinality()
3

center()#

Return the subgroup of elements that commute with every element of this group.

EXAMPLES:

sage: G = PermutationGroup([[(1,2,3,4)]])
sage: G.center()
Subgroup generated by [(1,2,3,4)] of
 (Permutation Group with generators [(1,2,3,4)])
sage: G = PermutationGroup([[(1,2,3,4)], [(1,2)]])
sage: G.center()
Subgroup generated by [()] of
 (Permutation Group with generators [(1,2), (1,2,3,4)])

centralizer(g)#

Return the centralizer of g in self.

EXAMPLES:

sage: G = PermutationGroup([[(1,2),(3,4)], [(1,2,3,4)]])
sage: g = G([(1,3)])
sage: G.centralizer(g)
Subgroup generated by [(2,4), (1,3)] of
 (Permutation Group with generators [(1,2)(3,4), (1,2,3,4)])
sage: g = G([(1,2,3,4)])
sage: G.centralizer(g)
Subgroup generated by [(1,2,3,4)] of
 (Permutation Group with generators [(1,2)(3,4), (1,2,3,4)])
sage: H = G.subgroup([G([(1,2,3,4)])])
sage: G.centralizer(H)
Subgroup generated by [(1,2,3,4)] of
 (Permutation Group with generators [(1,2)(3,4), (1,2,3,4)])

character(values)#

Return a group character from values, where values is a list of the values of the character evaluated on the conjugacy classes.

EXAMPLES:

sage: G = AlternatingGroup(4)
sage: n = len(G.conjugacy_classes_representatives())
sage: G.character([1]*n)                                                    # needs sage.rings.number_field
Character of Alternating group of order 4!/2 as a permutation group

character_table()#

Return the matrix of values of the irreducible characters of a permutation group \(G\) at the conjugacy classes of \(G\).

The columns represent the conjugacy classes of \(G\) and the rows represent the different irreducible characters in the ordering given by GAP.

EXAMPLES:

sage: G = PermutationGroup([[(1,2),(3,4)], [(1,2,3)]])
sage: G.order()
12
sage: G.character_table()                                                   # needs sage.rings.number_field
[         1          1          1          1]
[         1 -zeta3 - 1      zeta3          1]
[         1      zeta3 -zeta3 - 1          1]
[         3          0          0         -1]
sage: G = PermutationGroup([[(1,2),(3,4)], [(1,2,3)]])
sage: CT = gap(G).CharacterTable()

Type CT.Display() to display this nicely.

sage: G = PermutationGroup([[(1,2),(3,4)], [(1,2,3,4)]])
sage: G.order()
8
sage: G.character_table()                                                   # needs sage.rings.number_field
[ 1  1  1  1  1]
[ 1 -1 -1  1  1]
[ 1 -1  1 -1  1]
[ 1  1 -1 -1  1]
[ 2  0  0  0 -2]
sage: CT = gap(G).CharacterTable()

Again, type CT.Display() to display this nicely.

sage: # needs sage.rings.number_field
sage: SymmetricGroup(2).character_table()
[ 1 -1]
[ 1  1]
sage: SymmetricGroup(3).character_table()
[ 1 -1  1]
[ 2  0 -1]
[ 1  1  1]
sage: SymmetricGroup(5).character_table()
[ 1 -1  1  1 -1 -1  1]
[ 4 -2  0  1  1  0 -1]
[ 5 -1  1 -1 -1  1  0]
[ 6  0 -2  0  0  0  1]
[ 5  1  1 -1  1 -1  0]
[ 4  2  0  1 -1  0 -1]
[ 1  1  1  1  1  1  1]
sage: list(AlternatingGroup(6).character_table())
[(1, 1, 1, 1, 1, 1, 1), (5, 1, 2, -1, -1, 0, 0), (5, 1, -1, 2, -1, 0, 0),
 (8, 0, -1, -1, 0, zeta5^3 + zeta5^2 + 1, -zeta5^3 - zeta5^2),
 (8, 0, -1, -1, 0, -zeta5^3 - zeta5^2, zeta5^3 + zeta5^2 + 1),
 (9, 1, 0, 0, 1, -1, -1), (10, -2, 1, 1, 0, 0, 0)]

Suppose that you have a class function \(f(g)\) on \(G\) and you know the values \(v_1, \ldots, v_n\) on the conjugacy class elements in conjugacy_classes_representatives(G) = \([g_1, \ldots, g_n]\). Since the irreducible characters \(\rho_1, \ldots, \rho_n\) of \(G\) form an \(E\)-basis of the space of all class functions (\(E\) a “sufficiently large” cyclotomic field), such a class function is a linear combination of these basis elements, \(f = c_1 \rho_1 + \cdots + c_n \rho_n\). To find the coefficients \(c_i\), you simply solve the linear system character_table_values(G) \([v_1, ..., v_n] = [c_1, ..., c_n]\), where \([v_1, \ldots, v_n]\) = character_table_values(G) \(^{(-1)}[c_1, ..., c_n]\).

AUTHORS:

David Joyner and William Stein (2006-01-04)

cohomology(n, p=0)#

Computes the group cohomology \(H^n(G, F)\), where \(F = \ZZ\) if \(p=0\) and \(F = \ZZ / p \ZZ\) if \(p > 0\) is a prime.

Wraps HAP’s GroupHomology function, written by Graham Ellis.

REQUIRES: GAP package HAP (in gap_packages-*.spkg).

EXAMPLES:

sage: G = SymmetricGroup(4)
sage: G.cohomology(1,2)                            # optional - gap_package_hap
Multiplicative Abelian group isomorphic to C2
sage: G = SymmetricGroup(3)
sage: G.cohomology(5)                              # optional - gap_package_hap
Trivial Abelian group
sage: G.cohomology(5,2)                            # optional - gap_package_hap
Multiplicative Abelian group isomorphic to C2
sage: G.homology(5,3)                              # optional - gap_package_hap
Trivial Abelian group
sage: G.homology(5,4)                              # optional - gap_package_hap
Traceback (most recent call last):
...
ValueError: p must be 0 or prime

This computes \(H^4(S_3, \ZZ)\) and \(H^4(S_3, \ZZ / 2 \ZZ)\), respectively.

AUTHORS:

David Joyner and Graham Ellis

REFERENCES:

G. Ellis, ‘Computing group resolutions’, J. Symbolic Computation. Vol.38, (2004)1077-1118 (Available at http://hamilton.nuigalway.ie/).
D. Joyner, ‘A primer on computational group homology and cohomology’, http://front.math.ucdavis.edu/0706.0549.

cohomology_part(n, p=0)#

Compute the p-part of the group cohomology \(H^n(G, F)\), where \(F = \ZZ\) if \(p=0\) and \(F = \ZZ / p \ZZ\) if \(p > 0\) is a prime.

Wraps HAP’s Homology function, written by Graham Ellis, applied to the \(p\)-Sylow subgroup of \(G\).

REQUIRES: GAP package HAP (in gap_packages-*.spkg).

EXAMPLES:

sage: G = SymmetricGroup(5)
sage: G.cohomology_part(7,2)                   # optional - gap_package_hap
Multiplicative Abelian group isomorphic to C2 x C2 x C2
sage: G = SymmetricGroup(3)
sage: G.cohomology_part(2,3)                   # optional - gap_package_hap
Multiplicative Abelian group isomorphic to C3

AUTHORS:

David Joyner and Graham Ellis

commutator(other=None)#

Return the commutator subgroup of a group, or of a pair of groups.

INPUT:

other – (default: None) a permutation group.

OUTPUT:

Let \(G\) denote self. If other is None, then this method returns the subgroup of \(G\) generated by the set of commutators,

\[\{[g_1,g_2]\vert g_1, g_2\in G\} = \{g_1^{-1}g_2^{-1}g_1g_2\vert g_1, g_2\in G\}\]

Let \(H\) denote other, in the case that it is not None. Then this method returns the group generated by the set of commutators,

\[\{[g,h]\vert g\in G\, h\in H\} = \{g^{-1}h^{-1}gh\vert g\in G\, h\in H\}\]

The two groups need only be permutation groups, there is no notion of requiring them to explicitly be subgroups of some other group.

Note

For the identical statement, the generators of the returned group can vary from one execution to the next.

EXAMPLES:

sage: G = DiCyclicGroup(4)
sage: G.commutator()
Permutation Group with generators [(1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)]

sage: G = SymmetricGroup(5)
sage: H = CyclicPermutationGroup(5)
sage: C = G.commutator(H)
sage: C.is_isomorphic(AlternatingGroup(5))
True

An abelian group will have a trivial commutator.

sage: G = CyclicPermutationGroup(10)
sage: G.commutator()
Permutation Group with generators [()]

The quotient of a group by its commutator is always abelian.

sage: G = DihedralGroup(20)
sage: C = G.commutator()
sage: Q = G.quotient(C)
sage: Q.is_abelian()
True

When forming commutators from two groups, the order of the groups does not matter.

sage: D = DihedralGroup(3)
sage: S = SymmetricGroup(2)
sage: C1 = D.commutator(S); C1
Permutation Group with generators [(1,2,3)]
sage: C2 = S.commutator(D); C2
Permutation Group with generators [(1,3,2)]
sage: C1 == C2
True

This method calls two different functions in GAP, so this tests that their results are consistent. The commutator groups may have different generators, but the groups are equal.

sage: G = DiCyclicGroup(3)
sage: C = G.commutator(); C
Permutation Group with generators [(5,7,6)]
sage: CC = G.commutator(G); CC
Permutation Group with generators [(5,6,7)]
sage: C == CC
True

The second group is checked.

sage: G = SymmetricGroup(2)
sage: G.commutator('junk')
Traceback (most recent call last):
...
TypeError: junk is not a permutation group

composition_series()#

Return the composition series of this group as a list of permutation groups.

EXAMPLES:

These computations use pseudo-random numbers, so we set the seed for reproducible testing.

sage: set_random_seed(0)
sage: G = PermutationGroup([[(1,2,3),(4,5)],[(3,4)]])
sage: G.composition_series()
[Subgroup generated by [(3,4), (1,2,3)(4,5)] of
  (Permutation Group with generators [(3,4), (1,2,3)(4,5)]),
 Subgroup generated by [(1,3,5), (1,5)(3,4), (1,5)(2,4)] of
  (Permutation Group with generators [(3,4), (1,2,3)(4,5)]),
 Subgroup generated by [()] of
  (Permutation Group with generators [(3,4), (1,2,3)(4,5)])]
sage: G = PermutationGroup([[(1,2,3),(4,5)], [(1,2)]])
sage: CS = G.composition_series()
sage: CS[3]
Subgroup generated by [()] of
 (Permutation Group with generators [(1,2), (1,2,3)(4,5)])

conjugacy_class(g)#

Return the conjugacy class of g inside the group self.

INPUT:

g – an element of the permutation group self

OUTPUT:

The conjugacy class of g in the group self. If self is the group denoted by \(G\), this method computes the set \(\{x^{-1}gx\ \vert\ x \in G \}\)

EXAMPLES:

sage: G = DihedralGroup(3)
sage: g = G.gen(0)
sage: G.conjugacy_class(g)
Conjugacy class of (1,2,3) in Dihedral group of order 6 as a permutation group

conjugacy_classes()#

Return a list with all the conjugacy classes of self.

EXAMPLES:

sage: G = DihedralGroup(3)
sage: G.conjugacy_classes()
[Conjugacy class of () in Dihedral group of order 6 as a permutation group,
 Conjugacy class of (2,3) in Dihedral group of order 6 as a permutation group,
 Conjugacy class of (1,2,3) in Dihedral group of order 6 as a permutation group]

conjugacy_classes_representatives()#

Return a complete list of representatives of conjugacy classes in a permutation group \(G\).

The ordering is that given by GAP.

EXAMPLES:

sage: G = PermutationGroup([[(1,2),(3,4)], [(1,2,3,4)]])
sage: cl = G.conjugacy_classes_representatives(); cl
[(), (2,4), (1,2)(3,4), (1,2,3,4), (1,3)(2,4)]
sage: cl[3] in G
True

sage: G = SymmetricGroup(5)
sage: G.conjugacy_classes_representatives()                                 # needs sage.combinat
[(), (1,2), (1,2)(3,4), (1,2,3), (1,2,3)(4,5), (1,2,3,4), (1,2,3,4,5)]

sage: S = SymmetricGroup(['a','b','c'])
sage: S.conjugacy_classes_representatives()                                 # needs sage.combinat
[(), ('a','b'), ('a','b','c')]

AUTHORS:

David Joyner and William Stein (2006-01-04)

conjugacy_classes_subgroups()#

Return a complete list of representatives of conjugacy classes of subgroups in a permutation group \(G\).

The ordering is that given by GAP.

EXAMPLES:

sage: G = PermutationGroup([[(1,2),(3,4)], [(1,2,3,4)]])
sage: cl = G.conjugacy_classes_subgroups(); cl
[Subgroup generated by [()] of
  (Permutation Group with generators [(1,2)(3,4), (1,2,3,4)]),
 Subgroup generated by [(1,2)(3,4)] of
  (Permutation Group with generators [(1,2)(3,4), (1,2,3,4)]),
 Subgroup generated by [(1,3)(2,4)] of
  (Permutation Group with generators [(1,2)(3,4), (1,2,3,4)]),
 Subgroup generated by [(2,4)] of
  (Permutation Group with generators [(1,2)(3,4), (1,2,3,4)]),
 Subgroup generated by [(1,2)(3,4), (1,4)(2,3)] of
  (Permutation Group with generators [(1,2)(3,4), (1,2,3,4)]),
 Subgroup generated by [(2,4), (1,3)(2,4)] of
  (Permutation Group with generators [(1,2)(3,4), (1,2,3,4)]),
 Subgroup generated by [(1,2,3,4), (1,3)(2,4)] of
  (Permutation Group with generators [(1,2)(3,4), (1,2,3,4)]),
 Subgroup generated by [(2,4), (1,2)(3,4), (1,4)(2,3)] of
  (Permutation Group with generators [(1,2)(3,4), (1,2,3,4)])]

sage: G = SymmetricGroup(3)
sage: G.conjugacy_classes_subgroups()
[Subgroup generated by [()] of
  (Symmetric group of order 3! as a permutation group),
 Subgroup generated by [(2,3)] of
  (Symmetric group of order 3! as a permutation group),
 Subgroup generated by [(1,2,3)] of
  (Symmetric group of order 3! as a permutation group),
 Subgroup generated by [(2,3), (1,2,3)] of
  (Symmetric group of order 3! as a permutation group)]

AUTHORS:

David Joyner (2006-10)

conjugate(g)#

Return the group formed by conjugating self with g.

INPUT:

g – a permutation group element, or an object that converts to a permutation group element, such as a list of integers or a string of cycles.

OUTPUT:

If self is the group denoted by \(H\), then this method computes the group

\[g^{-1}Hg = \{g^{-1}hg\vert h\in H\}\]

which is the group \(H\) conjugated by \(g\).

There are no restrictions on self and g belonging to a common permutation group, and correspondingly, there is no relationship (such as a common parent) between self and the output group.

EXAMPLES:

sage: G = DihedralGroup(6)
sage: a = PermutationGroupElement("(1,2,3,4)")
sage: G.conjugate(a)
Permutation Group with generators [(1,4)(2,6)(3,5), (1,5,6,2,3,4)]

The element performing the conjugation can be specified in several ways.

sage: G = DihedralGroup(6)
sage: strng = "(1,2,3,4)"
sage: G.conjugate(strng)
Permutation Group with generators [(1,4)(2,6)(3,5), (1,5,6,2,3,4)]
sage: G = DihedralGroup(6)
sage: lst = [2,3,4,1]
sage: G.conjugate(lst)
Permutation Group with generators [(1,4)(2,6)(3,5), (1,5,6,2,3,4)]
sage: G = DihedralGroup(6)
sage: cycles = [(1,2,3,4)]
sage: G.conjugate(cycles)
Permutation Group with generators [(1,4)(2,6)(3,5), (1,5,6,2,3,4)]

Conjugation is a group automorphism, so conjugate groups will be isomorphic.

sage: G = DiCyclicGroup(6)
sage: G.degree()
11
sage: cycle = [i+1 for i in range(1,11)] + [1]
sage: C = G.conjugate(cycle)
sage: G.is_isomorphic(C)
True

The conjugating element may be from a symmetric group with larger degree than the group being conjugated.

sage: G = AlternatingGroup(5)
sage: G.degree()
5
sage: g = "(1,3)(5,6,7)"
sage: H = G.conjugate(g); H
Permutation Group with generators [(1,4,6,3,2), (1,4,6)]
sage: H.degree()
6

The conjugating element is checked.

sage: G = SymmetricGroup(3)
sage: G.conjugate("junk")
Traceback (most recent call last):
...
TypeError: junk does not convert to a permutation group element

construction()#

Return the construction of self.

EXAMPLES:

sage: P1 = PermutationGroup([[(1,2)]])
sage: P1.construction()
(PermutationGroupFunctor[(1,2)], Permutation Group with generators [()])

sage: PermutationGroup([]).construction() is None
True

This allows us to perform computations like the following:

sage: P1 = PermutationGroup([[(1,2)]]); p1 = P1.gen()
sage: P2 = PermutationGroup([[(1,3)]]); p2 = P2.gen()
sage: p = p1*p2; p
(1,2,3)
sage: p.parent()
Permutation Group with generators [(1,2), (1,3)]
sage: p.parent().domain()
{1, 2, 3}

Note that this will merge permutation groups with different domains:

sage: g1 = PermutationGroupElement([(1,2),(3,4,5)])
sage: g2 = PermutationGroup([('a','b')], domain=['a', 'b']).gens()[0]
sage: g2
('a','b')
sage: p = g1*g2; p
(1,2)(3,4,5)('a','b')
sage: P = parent(p)
sage: P
Permutation Group with generators [('a','b'), (1,2), (1,2,3,4,5)]

cosets(S, side='right')#

Return a list of the cosets of S in self.

INPUT:

S – a subgroup of self. An error is raised if S is not a subgroup.
side – (default: 'right') Determines if right cosets or left cosets are returned. side refers to where the representative is placed in the products forming the cosets and thus allowable values are only 'right' and 'left'.

OUTPUT:

A list of lists. Each inner list is a coset of the subgroup in the group. The first element of each coset is the smallest element (based on the ordering of the elements of self) of all the group elements that have not yet appeared in a previous coset. The elements of each coset are in the same order as the subgroup elements used to build the coset’s elements.

As a consequence, the subgroup itself is the first coset, and its first element is the identity element. For each coset, the first element listed is the element used as a representative to build the coset. These representatives form an increasing sequence across the list of cosets, and within a coset the representative is the smallest element of its coset (both orderings are based on of the ordering of elements of self).

In the case of a normal subgroup, left and right cosets should appear in the same order as part of the outer list. However, the list of the elements of a particular coset may be in a different order for the right coset versus the order in the left coset. So, if you check to see if a subgroup is normal, it is necessary to sort each individual coset first (but not the list of cosets, due to the ordering of the representatives). See below for examples of this.

Note

This is a naive implementation intended for instructional purposes, and hence is slow for larger groups. Sage and GAP provide more sophisticated functions for working quickly with cosets of larger groups.

EXAMPLES:

The default is to build right cosets. This example works with the symmetry group of an 8-gon and a normal subgroup. Notice that a straight check on the equality of the output is not sufficient to check normality, while sorting the individual cosets is sufficient to then simply test equality of the list of lists. Study the second coset in each list to understand the need for sorting the elements of the cosets.

sage: G = DihedralGroup(8)
sage: quarter_turn = G('(1,3,5,7)(2,4,6,8)'); quarter_turn
(1,3,5,7)(2,4,6,8)
sage: S = G.subgroup([quarter_turn])
sage: rc = G.cosets(S); rc
[[(), (1,3,5,7)(2,4,6,8), (1,5)(2,6)(3,7)(4,8), (1,7,5,3)(2,8,6,4)],
 [(2,8)(3,7)(4,6), (1,7)(2,6)(3,5), (1,5)(2,4)(6,8), (1,3)(4,8)(5,7)],
 [(1,2)(3,8)(4,7)(5,6), (1,8)(2,7)(3,6)(4,5), (1,6)(2,5)(3,4)(7,8), (1,4)(2,3)(5,8)(6,7)],
 [(1,2,3,4,5,6,7,8), (1,4,7,2,5,8,3,6), (1,6,3,8,5,2,7,4), (1,8,7,6,5,4,3,2)]]
sage: lc = G.cosets(S, side='left'); lc
[[(), (1,3,5,7)(2,4,6,8), (1,5)(2,6)(3,7)(4,8), (1,7,5,3)(2,8,6,4)],
 [(2,8)(3,7)(4,6), (1,3)(4,8)(5,7), (1,5)(2,4)(6,8), (1,7)(2,6)(3,5)],
 [(1,2)(3,8)(4,7)(5,6), (1,4)(2,3)(5,8)(6,7), (1,6)(2,5)(3,4)(7,8), (1,8)(2,7)(3,6)(4,5)],
 [(1,2,3,4,5,6,7,8), (1,4,7,2,5,8,3,6), (1,6,3,8,5,2,7,4), (1,8,7,6,5,4,3,2)]]

sage: S.is_normal(G)
True
sage: rc == lc
False
sage: rc_sorted = [sorted(c) for c in rc]
sage: lc_sorted = [sorted(c) for c in lc]
sage: rc_sorted == lc_sorted
True

An example with the symmetry group of a regular tetrahedron and a subgroup that is not normal. Thus, the right and left cosets are different (and so are the representatives). With each individual coset sorted, a naive test of normality is possible.

sage: A = AlternatingGroup(4)
sage: face_turn = A('(1,2,3)'); face_turn
(1,2,3)
sage: stabilizer = A.subgroup([face_turn])
sage: rc = A.cosets(stabilizer, side='right'); rc
[[(), (1,2,3), (1,3,2)],
 [(2,3,4), (1,3)(2,4), (1,4,2)],
 [(2,4,3), (1,4,3), (1,2)(3,4)],
 [(1,2,4), (1,4)(2,3), (1,3,4)]]
sage: lc = A.cosets(stabilizer, side='left'); lc
[[(), (1,2,3), (1,3,2)],
 [(2,3,4), (1,2)(3,4), (1,3,4)],
 [(2,4,3), (1,2,4), (1,3)(2,4)],
 [(1,4,2), (1,4,3), (1,4)(2,3)]]

sage: stabilizer.is_normal(A)
False
sage: rc_sorted = [sorted(c) for c in rc]
sage: lc_sorted = [sorted(c) for c in lc]
sage: rc_sorted == lc_sorted
False

AUTHOR:

Rob Beezer (2011-01-31)

degree()#

Return the degree of this permutation group.

EXAMPLES:

sage: S = SymmetricGroup(['a','b','c'])
sage: S.degree()
3
sage: G = PermutationGroup([(1,3),(4,5)])
sage: G.degree()
5

Note that you can explicitly specify the domain to get a permutation group of smaller degree:

sage: G = PermutationGroup([(1,3),(4,5)], domain=[1,3,4,5])
sage: G.degree()
4

derived_series()#

Return the derived series of this group as a list of permutation groups.

EXAMPLES:

These computations use pseudo-random numbers, so we set the seed for reproducible testing.

sage: set_random_seed(0)
sage: G = PermutationGroup([[(1,2,3),(4,5)],[(3,4)]])
sage: G.derived_series()
[Subgroup generated by [(3,4), (1,2,3)(4,5)] of
  (Permutation Group with generators [(3,4), (1,2,3)(4,5)]),
 Subgroup generated by [(1,3,5), (1,5)(3,4), (1,5)(2,4)] of
  (Permutation Group with generators [(3,4), (1,2,3)(4,5)])]

direct_product(other, maps=True)#

Wraps GAP’s DirectProduct, Embedding, and Projection.

Sage calls GAP’s DirectProduct, which chooses an efficient representation for the direct product. The direct product of permutation groups will be a permutation group again. For a direct product D, the GAP operation Embedding(D,i) returns the homomorphism embedding the i-th factor into D. The GAP operation Projection(D,i) gives the projection of D onto the i-th factor. This method returns a 5-tuple: a permutation group and 4 morphisms.

INPUT:

self, other – permutation groups

OUTPUT:

D – a direct product of the inputs, returned as a permutation group as well
iota1 – an embedding of self into D
iota2 – an embedding of other into D
pr1 – the projection of D onto self (giving a splitting 1 - other - D - self - 1)
pr2 – the projection of D onto other (giving a splitting 1 - self - D - other - 1)

EXAMPLES:

sage: G = CyclicPermutationGroup(4)
sage: D = G.direct_product(G, False); D
Permutation Group with generators [(5,6,7,8), (1,2,3,4)]
sage: D,iota1,iota2,pr1,pr2 = G.direct_product(G)
sage: D; iota1; iota2; pr1; pr2
Permutation Group with generators [(5,6,7,8), (1,2,3,4)]
Permutation group morphism:
  From: Cyclic group of order 4 as a permutation group
  To:   Permutation Group with generators [(5,6,7,8), (1,2,3,4)]
  Defn: Embedding( Group( [ (1,2,3,4), (5,6,7,8) ] ), 1 )
Permutation group morphism:
  From: Cyclic group of order 4 as a permutation group
  To:   Permutation Group with generators [(5,6,7,8), (1,2,3,4)]
  Defn: Embedding( Group( [ (1,2,3,4), (5,6,7,8) ] ), 2 )
Permutation group morphism:
  From: Permutation Group with generators [(5,6,7,8), (1,2,3,4)]
  To:   Cyclic group of order 4 as a permutation group
  Defn: Projection( Group( [ (1,2,3,4), (5,6,7,8) ] ), 1 )
Permutation group morphism:
  From: Permutation Group with generators [(5,6,7,8), (1,2,3,4)]
  To:   Cyclic group of order 4 as a permutation group
  Defn: Projection( Group( [ (1,2,3,4), (5,6,7,8) ] ), 2 )
sage: g = D([(1,3),(2,4)]); g
(1,3)(2,4)
sage: d = D([(1,4,3,2),(5,7),(6,8)]); d
(1,4,3,2)(5,7)(6,8)
sage: iota1(g); iota2(g); pr1(d); pr2(d)
(1,3)(2,4)
(5,7)(6,8)
(1,4,3,2)
(1,3)(2,4)

domain()#

Return the underlying set that this permutation group acts on.

EXAMPLES:

sage: P = PermutationGroup([(1,2),(3,5)])
sage: P.domain()
{1, 2, 3, 4, 5}
sage: S = SymmetricGroup(['a', 'b', 'c'])
sage: S.domain()
{'a', 'b', 'c'}

exponent()#

Computes the exponent of the group.

The exponent \(e\) of a group \(G\) is the LCM of the orders of its elements, that is, \(e\) is the smallest integer such that \(g^e=1\) for all \(g \in G\).

EXAMPLES:

sage: G = AlternatingGroup(4)
sage: G.exponent()
6

fitting_subgroup()#

Return the Fitting subgroup of self.

The Fitting subgroup of a group \(G\) is the largest nilpotent normal subgroup of \(G\).

EXAMPLES:

sage: G = PermutationGroup([[(1,2,3,4)],[(2,4)]])
sage: G.fitting_subgroup()
Subgroup generated by [(2,4), (1,2,3,4), (1,3)] of
 (Permutation Group with generators [(2,4), (1,2,3,4)])
sage: G = PermutationGroup([[(1,2,3,4)],[(1,2)]])
sage: G.fitting_subgroup()
Subgroup generated by [(1,2)(3,4), (1,3)(2,4)] of
 (Permutation Group with generators [(1,2), (1,2,3,4)])

fixed_points()#

Return the list of points fixed by self, i.e., the subset of .domain() not moved by any element of self.

EXAMPLES:

sage: G = PermutationGroup([(1,2,3)])
sage: G.fixed_points()
[]
sage: G = PermutationGroup([(1,2,3),(5,6)])
sage: G.fixed_points()
[4]
sage: G = PermutationGroup([[(1,4,7)],[(4,3),(6,7)]])
sage: G.fixed_points()
[2, 5]

frattini_subgroup()#

Return the Frattini subgroup of self.

The Frattini subgroup of a group \(G\) is the intersection of all maximal subgroups of \(G\).

EXAMPLES:

sage: G = PermutationGroup([[(1,2,3,4)],[(2,4)]])
sage: G.frattini_subgroup()
Subgroup generated by [(1,3)(2,4)] of
 (Permutation Group with generators [(2,4), (1,2,3,4)])
sage: G = SymmetricGroup(4)
sage: G.frattini_subgroup()
Subgroup generated by [()] of
 (Symmetric group of order 4! as a permutation group)

gap()#

This method from sage.groups.libgap_wrapper.ParentLibGAP is added in order to achieve compatibility and have sage.groups.libgap_morphism.GroupHomset_libgap work for permutation groups, as well

OUTPUT:

an instance of sage.libs.gap.element.GapElement representing this group

EXAMPLES:

sage: P8 = PSp(8,3)
sage: P8.gap()
<permutation group of size 65784756654489600 with 2 generators>
sage: gap(P8) == P8.gap()
False
sage: S3 = SymmetricGroup(3)
sage: S3.gap()
Sym( [ 1 .. 3 ] )
sage: gap(S3) == S3.gap()
False

gen(i=None)#

Return the \(i\)-th generator of self; that is, the \(i\)-th element of the list self.gens().

The argument \(i\) may be omitted if there is only one generator (but this will raise an error otherwise).

EXAMPLES:

We explicitly construct the alternating group on four elements:

sage: A4 = PermutationGroup([[(1,2,3)],[(2,3,4)]]); A4
Permutation Group with generators [(2,3,4), (1,2,3)]
sage: A4.gens()
((2,3,4), (1,2,3))
sage: A4.gen(0)
(2,3,4)
sage: A4.gen(1)
(1,2,3)
sage: A4.gens()[0]; A4.gens()[1]
(2,3,4)
(1,2,3)

sage: P1 = PermutationGroup([[(1,2)]]); P1.gen()
(1,2)

gens()#

Return tuple of generators of this group.

These need not be minimal, as they are the generators used in defining this group.

EXAMPLES:

sage: G = PermutationGroup([[(1,2,3)], [(1,2)]])
sage: G.gens()
((1,2), (1,2,3))

Note that the generators need not be minimal, though duplicates are removed:

sage: G = PermutationGroup([[(1,2)], [(1,3)], [(2,3)], [(1,2)]])
sage: G.gens()
((2,3), (1,2), (1,3))

We can use index notation to access the generators returned by self.gens:

sage: G = PermutationGroup([[(1,2,3,4), (5,6)], [(1,2)]])
sage: g = G.gens()
sage: g[0]
(1,2)
sage: g[1]
(1,2,3,4)(5,6)

gens_small()#

For this group, returns a generating set which has few elements.

As neither irredundancy nor minimal length is proven, it is fast.

EXAMPLES:

sage: R = "(25,27,32,30)(26,29,31,28)( 3,38,43,19)( 5,36,45,21)( 8,33,48,24)" ## R = right
sage: U = "( 1, 3, 8, 6)( 2, 5, 7, 4)( 9,33,25,17)(10,34,26,18)(11,35,27,19)" ## U = top
sage: L = "( 9,11,16,14)(10,13,15,12)( 1,17,41,40)( 4,20,44,37)( 6,22,46,35)" ## L = left
sage: F = "(17,19,24,22)(18,21,23,20)( 6,25,43,16)( 7,28,42,13)( 8,30,41,11)" ## F = front
sage: B = "(33,35,40,38)(34,37,39,36)( 3, 9,46,32)( 2,12,47,29)( 1,14,48,27)" ## B = back or rear
sage: D = "(41,43,48,46)(42,45,47,44)(14,22,30,38)(15,23,31,39)(16,24,32,40)" ## D = down or bottom
sage: G = PermutationGroup([R,L,U,F,B,D])
sage: len(G.gens_small())
2

The output may be unpredictable, due to the use of randomized algorithms in GAP. Note that both the following answers are equally valid.

sage: G = PermutationGroup([[('a','b')], [('b', 'c')], [('a', 'c')]])
sage: G.gens_small() # random
[('b','c'), ('a','c','b')] ## (on 64-bit Linux)
[('a','b'), ('a','c','b')] ## (on Solaris)
sage: len(G.gens_small()) == 2 # random
True

group_id()#

Return the ID code of this group, which is a list of two integers.

EXAMPLES:

sage: G = PermutationGroup([[(1,2,3),(4,5)], [(1,2)]])
sage: G.group_id()
[12, 4]

group_primitive_id()#

Return the index of this group in the GAP database of primitive groups.

OUTPUT:

A positive integer, following GAP’s conventions. A ValueError is raised if the group is not primitive.

EXAMPLES:

sage: G = PermutationGroup([[(1,2,3,4,5)], [(1,5),(2,4)]])
sage: G.group_primitive_id()
2
sage: G.degree()
5

From the information of the degree and the identification number, you can recover the isomorphism class of your group in the GAP database:

sage: H = PrimitiveGroup(5,2)
sage: G == H
False
sage: G.is_isomorphic(H)
True

has_element(item)#

Return whether item is an element of this group - however ignores parentage.

EXAMPLES:

sage: G = CyclicPermutationGroup(4)
sage: gens = G.gens()
sage: H = DihedralGroup(4)
sage: g = G([(1,2,3,4)]); g
(1,2,3,4)
sage: G.has_element(g)
doctest:warning
...
DeprecationWarning: G.has_element(g) is deprecated; use :meth:`__contains__`, i.e., `g in G` instead
See https://github.com/sagemath/sage/issues/33831 for details.
True
sage: h = H([(1,2),(3,4)]); h
(1,2)(3,4)
sage: G.has_element(h)
False

has_regular_subgroup(return_group=False)#

Return whether the group contains a regular subgroup.

INPUT:

return_group – (boolean) If True, a regular subgroup is returned if there is one, and None if there isn’t. When return_group=False (default), only a boolean indicating whether such a group exists is returned instead.

EXAMPLES:

The symmetric group on 4 elements has a regular subgroup:

sage: S4 = groups.permutation.Symmetric(4)
sage: S4.has_regular_subgroup()
True
sage: S4.has_regular_subgroup(return_group=True)  # random
Subgroup of (Symmetric group of order 4! as a permutation group)
 generated by [(1,3)(2,4), (1,4)(2,3)]

But the automorphism group of Petersen’s graph does not:

sage: # needs sage.graphs
sage: G = graphs.PetersenGraph().automorphism_group()
sage: G.has_regular_subgroup()
False

holomorph()#

The holomorph of a group as a permutation group.

The holomorph of a group \(G\) is the semidirect product \(G \rtimes_{id} Aut(G)\), where \(id\) is the identity function on \(Aut(G)\), the automorphism group of \(G\).

See Wikipedia article Holomorph (mathematics)

OUTPUT:

Return the holomorph of a given group as permutation group via a wrapping of GAP’s semidirect product function.

EXAMPLES:

Thomas and Wood’s ‘Group Tables’ (Shiva Publishing, 1980) tells us that the holomorph of \(C_5\) is the unique group of order 20 with a trivial center.

sage: C5 = CyclicPermutationGroup(5)
sage: A = C5.holomorph()
sage: A.order()
20
sage: A.is_abelian()
False
sage: A.center()
Subgroup generated by [()] of
 (Permutation Group with generators [(5,6,7,8,9), (1,2,4,3)(6,7,9,8)])
sage: A
Permutation Group with generators [(5,6,7,8,9), (1,2,4,3)(6,7,9,8)]

Noting that the automorphism group of \(D_4\) is itself \(D_4\), it can easily be shown that the holomorph is indeed an internal semidirect product of these two groups.

sage: D4 = DihedralGroup(4)
sage: H = D4.holomorph()
sage: H.gens()
((3,8)(4,7), (2,3,5,8), (2,5)(3,8), (1,4,6,7)(2,3,5,8), (1,8)(2,7)(3,6)(4,5))
sage: G = H.subgroup([H.gens()[0],H.gens()[1],H.gens()[2]])
sage: N = H.subgroup([H.gens()[3],H.gens()[4]])
sage: N.is_normal(H)
True
sage: G.is_isomorphic(D4)
True
sage: N.is_isomorphic(D4)
True
sage: G.intersection(N)
Permutation Group with generators [()]
sage: L = [H(x)*H(y) for x in G for y in N]; L.sort()
sage: L1 = H.list(); L1.sort()
sage: L == L1
True

Author:

Kevin Halasz (2012-08-14)

homology(n, p=0)#

Computes the group homology \(H_n(G, F)\), where \(F = \ZZ\) if \(p=0\) and \(F = \ZZ / p \ZZ\) if \(p > 0\) is a prime. Wraps HAP’s GroupHomology function, written by Graham Ellis.

REQUIRES: GAP package HAP (in gap_packages-*.spkg).

AUTHORS:

David Joyner and Graham Ellis

The example below computes \(H_7(S_5, \ZZ)\), \(H_7(S_5, \ZZ / 2 \ZZ)\), \(H_7(S_5, \ZZ / 3 \ZZ)\), and \(H_7(S_5, \ZZ / 5 \ZZ)\), respectively. To compute the \(2\)-part of \(H_7(S_5, \ZZ)\), use the method homology_part().

EXAMPLES:

sage: G = SymmetricGroup(5)
sage: G.homology(7)                              # optional - gap_package_hap
Multiplicative Abelian group isomorphic to C2 x C2 x C4 x C3 x C5
sage: G.homology(7,2)                              # optional - gap_package_hap
Multiplicative Abelian group isomorphic to C2 x C2 x C2 x C2 x C2
sage: G.homology(7,3)                              # optional - gap_package_hap
Multiplicative Abelian group isomorphic to C3
sage: G.homology(7,5)                              # optional - gap_package_hap
Multiplicative Abelian group isomorphic to C5

REFERENCES:

G. Ellis, “Computing group resolutions”, J. Symbolic Computation. Vol.38, (2004)1077-1118 (Available at http://hamilton.nuigalway.ie/.
D. Joyner, “A primer on computational group homology and cohomology”, http://front.math.ucdavis.edu/0706.0549

homology_part(n, p=0)#

Computes the \(p\)-part of the group homology \(H_n(G, F)\), where \(F = \ZZ\) if \(p=0\) and \(F = \ZZ / p \ZZ\) if \(p > 0\) is a prime. Wraps HAP’s Homology function, written by Graham Ellis, applied to the \(p\)-Sylow subgroup of \(G\).

REQUIRES: GAP package HAP (in gap_packages-*.spkg).

EXAMPLES:

sage: G = SymmetricGroup(5)
sage: G.homology_part(7,2)                              # optional - gap_package_hap
Multiplicative Abelian group isomorphic to C2 x C2 x C2 x C2 x C4

AUTHORS:

David Joyner and Graham Ellis

id()#

Return the ID code of this group, which is a list of two integers.

Same as group_id().

EXAMPLES:

sage: G = PermutationGroup([[(1,2,3),(4,5)], [(1,2)]])
sage: G.group_id()
[12, 4]

identity()#

Return the identity element of this group.

EXAMPLES:

sage: G = PermutationGroup([[(1,2,3),(4,5)]])
sage: e = G.identity(); e                       # indirect doctest
()
sage: g = G.gen(0)
sage: g*e
(1,2,3)(4,5)
sage: e*g
(1,2,3)(4,5)

sage: S = SymmetricGroup(['a','b','c'])
sage: S.identity()
()

intersection(other)#

Return the permutation group that is the intersection of self and other.

INPUT:

other - a permutation group.

OUTPUT:

A permutation group that is the set-theoretic intersection of self with other. The groups are viewed as subgroups of a symmetric group big enough to contain both group’s symbol sets. So there is no strict notion of the two groups being subgroups of a common parent.

EXAMPLES:

sage: H = DihedralGroup(4)

sage: K = CyclicPermutationGroup(4)
sage: H.intersection(K)
Permutation Group with generators [(1,2,3,4)]

sage: L = DihedralGroup(5)
sage: H.intersection(L)
Permutation Group with generators [(1,4)(2,3)]

sage: M = PermutationGroup(["()"])
sage: H.intersection(M)
Permutation Group with generators [()]

Some basic properties.

sage: H = DihedralGroup(4)
sage: L = DihedralGroup(5)
sage: H.intersection(L) == L.intersection(H)
True
sage: H.intersection(H) == H
True

The group other is verified as such.

sage: H = DihedralGroup(4)
sage: H.intersection('junk')
Traceback (most recent call last):
...
TypeError: junk is not a permutation group

irreducible_characters()#

Return a list of the irreducible characters of self.

EXAMPLES:

sage: irr = SymmetricGroup(3).irreducible_characters()                      # needs sage.rings.number_field
sage: [x.values() for x in irr]                                             # needs sage.rings.number_field
[[1, -1, 1], [2, 0, -1], [1, 1, 1]]

is_abelian()#

Return True if this group is abelian.

EXAMPLES:

sage: G = PermutationGroup(['(1,2,3)(4,5)', '(1,2,3,4,5)'])
sage: G.is_abelian()
False
sage: G = PermutationGroup(['(1,2,3)(4,5)'])
sage: G.is_abelian()
True

is_commutative()#

Return True if this group is commutative.

EXAMPLES:

sage: G = PermutationGroup(['(1,2,3)(4,5)', '(1,2,3,4,5)'])
sage: G.is_commutative()
False
sage: G = PermutationGroup(['(1,2,3)(4,5)'])
sage: G.is_commutative()
True

is_cyclic()#

Return True if this group is cyclic.

EXAMPLES:

sage: G = PermutationGroup(['(1,2,3)(4,5)', '(1,2,3,4,5)'])
sage: G.is_cyclic()
False
sage: G = PermutationGroup(['(1,2,3)(4,5)'])
sage: G.is_cyclic()
True

is_elementary_abelian()#

Return True if this group is elementary abelian. An elementary abelian group is a finite abelian group, where every nontrivial element has order \(p\), where \(p\) is a prime.

EXAMPLES:

sage: G = PermutationGroup(['(1,2,3)(4,5)', '(1,2,3,4,5)'])
sage: G.is_elementary_abelian()
False
sage: G = PermutationGroup(['(1,2,3)','(4,5,6)'])
sage: G.is_elementary_abelian()
True

is_isomorphic(right)#

Return True if the groups are isomorphic.

INPUT:

self – this group
right – a permutation group

OUTPUT:

boolean; True if self and right are isomorphic groups; False otherwise.

EXAMPLES:

sage: v = ['(1,2,3)(4,5)', '(1,2,3,4,5)']
sage: G = PermutationGroup(v)
sage: H = PermutationGroup(['(1,2,3)(4,5)'])
sage: G.is_isomorphic(H)
False
sage: G.is_isomorphic(G)
True
sage: G.is_isomorphic(PermutationGroup(list(reversed(v))))
True

is_monomial()#

Return True if the group is monomial. A finite group is monomial if every irreducible complex character is induced from a linear character of a subgroup.

EXAMPLES:

sage: G = PermutationGroup(['(1,2,3)(4,5)'])
sage: G.is_monomial()
True

is_nilpotent()#

Return True if this group is nilpotent.

EXAMPLES:

sage: G = PermutationGroup(['(1,2,3)(4,5)', '(1,2,3,4,5)'])
sage: G.is_nilpotent()
False
sage: G = PermutationGroup(['(1,2,3)(4,5)'])
sage: G.is_nilpotent()
True

is_normal(other)#

Return True if this group is a normal subgroup of other.

EXAMPLES:

sage: AlternatingGroup(4).is_normal(SymmetricGroup(4))
True
sage: H = PermutationGroup(['(1,2,3)(4,5)'])
sage: G = PermutationGroup(['(1,2,3)(4,5)', '(1,2,3,4,5)'])
sage: H.is_normal(G)
False

is_perfect()#

Return True if this group is perfect. A group is perfect if it equals its derived subgroup.

EXAMPLES:

sage: G = PermutationGroup(['(1,2,3)(4,5)', '(1,2,3,4,5)'])
sage: G.is_perfect()
False
sage: G = PermutationGroup(['(1,2,3)(4,5)'])
sage: G.is_perfect()
False

is_pgroup()#

Return True if this group is a \(p\)-group.

A finite group is a \(p\)-group if its order is of the form \(p^n\) for a prime integer \(p\) and a nonnegative integer \(n\).

EXAMPLES:

sage: G = PermutationGroup(['(1,2,3,4,5)'])
sage: G.is_pgroup()
True

is_polycyclic()#

Return True if this group is polycyclic. A group is polycyclic if it has a subnormal series with cyclic factors. (For finite groups, this is the same as if the group is solvable - see is_solvable().)

EXAMPLES:

sage: G = PermutationGroup(['(1,2,3)(4,5)', '(1,2,3,4,5)'])
sage: G.is_polycyclic()
False
sage: G = PermutationGroup(['(1,2,3)(4,5)'])
sage: G.is_polycyclic()
True

is_primitive(domain=None)#

Return True if self acts primitively on domain.

A group \(G\) acts primitively on a set \(S\) if

\(G\) acts transitively on \(S\) and
the action induces no non-trivial block system on \(S\).

INPUT:

domain (optional)

See also: degree().

EXAMPLES:

sage: G = PermutationGroup([[(1,2,3),(4,5)], [(1,2)]])
sage: G.order()
12
sage: G = PermutationGroup([()])
sage: G.order()
1
sage: G = PermutationGroup([])
sage: G.order()
1

cardinality() is just an alias:

sage: PermutationGroup([(1,2,3)]).cardinality()
3

poincare_series(p=2, n=10)#

Return the Poincaré series of \(G \mod p\) (\(p \geq 2\) must be a prime), for \(n\) large.

In other words, if you input a finite group \(G\), a prime \(p\), and a positive integer \(n\), it returns a quotient of polynomials \(f(x) = P(x) / Q(x)\) whose coefficient of \(x^k\) equals the rank of the vector space \(H_k(G, \ZZ / p \ZZ)\), for all \(k\) in the range \(1 \leq k \leq n\).

REQUIRES: GAP package HAP (in gap_packages-*.spkg).

EXAMPLES:

sage: G = SymmetricGroup(5)
sage: G.poincare_series(2, 10)                             # optional - gap_package_hap
(x^2 + 1)/(x^4 - x^3 - x + 1)
sage: G = SymmetricGroup(3)
sage: G.poincare_series(2, 10)                             # optional - gap_package_hap
-1/(x - 1)

AUTHORS:

David Joyner and Graham Ellis

quotient(N, **kwds)#

Return the quotient of this permutation group by the normal subgroup \(N\), as a permutation group.

Further named arguments are passed to the permutation group constructor.

Wraps the GAP operator “/”.

EXAMPLES:

sage: G = PermutationGroup([(1,2,3), (2,3)])
sage: N = PermutationGroup([(1,2,3)])
sage: G.quotient(N)
Permutation Group with generators [(1,2)]
sage: G.quotient(G)
Permutation Group with generators [()]

random_element()#

Return a random element of this group.

EXAMPLES:

sage: G = PermutationGroup([[(1,2,3),(4,5)], [(1,2)]])
sage: a = G.random_element()
sage: a in G
True
sage: a.parent() is G
True
sage: a^6
()

representative_action(x, y)#

Return an element of self that maps \(x\) to \(y\) if it exists.

This method wraps the gap function RepresentativeAction, which can also return elements that map a given set of points on another set of points.

INPUT:

x, y – two elements of the domain.

EXAMPLES:

sage: G = groups.permutation.Cyclic(14)
sage: g = G.representative_action(1, 10)
sage: all(g(x) == 1 + ((x+9-1)%14) for x in G.domain())
True

semidirect_product(N, mapping, check=True)#

The semidirect product of self with N.

INPUT:

N - A group which is acted on by self and naturally embeds as a normal subgroup of the returned semidirect product.
mapping – A pair of lists that together define a homomorphism, \(\phi :\) self \(\rightarrow\) Aut(N), by giving, in the second list, the images of the generators of self in the order given in the first list.
check – A boolean that, if set to False, will skip the initial tests which are made on mapping. This may be beneficial for large N, since in such cases the injectivity test can be expensive. Set to True by default.

OUTPUT:

The semidirect product of self and N defined by the action of self on N given in mapping (note that a homomorphism from \(A\) to the automorphism group of \(B\) is equivalent to an action of \(A\) on \(B\)’s underlying set). The semidirect product of two groups, \(H\) and \(N\), is a construct similar to the direct product in so far as the elements are the Cartesian product of the elements of \(H\) and the elements of \(N\). The operation, however, is built upon an action of \(H\) on \(N\), and is defined as such:

\[(h_1,n_1)(h_2,n_2) = (h_{1}h_{2}, n_{1}^{h_2}n_2)\]

This function is a wrapper for GAP’s SemidirectProduct command. The permutation group returned is built upon a permutation representation of the semidirect product of self and N on a set of size \(\mid N \mid\). The generators of N are given as their right regular representations, while the generators of self are defined by the underlying action of self on N. It should be noted that the defining action is not always faithful, and in this case the inputted representations of the generators of self are placed on additional letters and adjoined to the output’s generators of self.

EXAMPLES:

Perhaps the most common example of a semidirect product comes from the family of dihedral groups. Each dihedral group is the semidirect product of \(C_2\) with \(C_n\), where, by convention, \(3 \leq n\). In this case, the nontrivial element of \(C_2\) acts on \(C_n\) so as to send each element to its inverse.

sage: C2 = CyclicPermutationGroup(2)
sage: C8 = CyclicPermutationGroup(8)
sage: alpha = PermutationGroupMorphism_im_gens(C8,C8,[(1,8,7,6,5,4,3,2)])
sage: S = C2.semidirect_product(C8,[[(1,2)],[alpha]])
sage: S == DihedralGroup(8)
False
sage: S.is_isomorphic(DihedralGroup(8))
True
sage: S.gens()
((3,4,5,6,7,8,9,10), (1,2)(4,10)(5,9)(6,8))

A more complicated example can be drawn from [TW1980]. It is there given that a semidirect product of \(D_4\) and \(C_3\) is isomorphic to one of \(C_2\) and the dicyclic group of order 12. This nonabelian group of order 24 has very similar structure to the dicyclic and dihedral groups of order 24, the three being the only groups of order 24 with a two-element center and 9 conjugacy classes.

sage: D4 = DihedralGroup(4)
sage: C3 = CyclicPermutationGroup(3)
sage: alpha1 = PermutationGroupMorphism_im_gens(C3,C3,[(1,3,2)])
sage: alpha2 = PermutationGroupMorphism_im_gens(C3,C3,[(1,2,3)])
sage: S1 = D4.semidirect_product(C3,[[(1,2,3,4),(1,3)],[alpha1,alpha2]])
sage: C2 = CyclicPermutationGroup(2)
sage: Q = DiCyclicGroup(3)
sage: a = Q.gens()[0]; b=Q.gens()[1].inverse()
sage: alpha = PermutationGroupMorphism_im_gens(Q,Q,[a,b])
sage: S2 = C2.semidirect_product(Q,[[(1,2)],[alpha]])
sage: S1.is_isomorphic(S2)
True
sage: S1.is_isomorphic(DihedralGroup(12))
False
sage: S1.is_isomorphic(DiCyclicGroup(6))
False
sage: S1.center()
Subgroup generated by [(1,3)(2,4)] of
 (Permutation Group with generators [(5,6,7), (1,2,3,4)(6,7), (1,3)])
sage: len(S1.conjugacy_classes_representatives())
9

If your normal subgroup is large, and you are confident that your inputs will successfully create a semidirect product, then it is beneficial, for the sake of time efficiency, to set the check parameter to False.

sage: C2 = CyclicPermutationGroup(2)
sage: C2000 = CyclicPermutationGroup(500)
sage: alpha = PermutationGroupMorphism(C2000,C2000,[C2000.gen().inverse()])
sage: S = C2.semidirect_product(C2000,[[(1,2)],[alpha]],check=False)

AUTHOR:

Kevin Halasz (2012-8-12)

sign_representation(base_ring=None, side='twosided')#

Return the sign representation of self over base_ring.

INPUT:

base_ring – (optional) the base ring; the default is \(\ZZ\)
side – ignored

EXAMPLES:

sage: G = groups.permutation.Dihedral(4)
sage: G.sign_representation()
Sign representation of Dihedral group of order 8
 as a permutation group over Integer Ring

smallest_moved_point()#

Return the smallest point moved by a permutation in this group.

EXAMPLES:

sage: G = PermutationGroup([[(3,4)], [(2,3,4)]])
sage: G.smallest_moved_point()
2
sage: G = PermutationGroup([[(1,2),(3,4)], [(1,2,3,4,10)]])
sage: G.smallest_moved_point()
1

Note that this function uses the ordering from the domain:

sage: S = SymmetricGroup(['a','b','c'])
sage: S.smallest_moved_point()
'a'

socle()#

Return the socle of self.

The socle of a group \(G\) is the subgroup generated by all minimal normal subgroups.

EXAMPLES:

sage: G = SymmetricGroup(4)
sage: s = G.socle(); s
Subgroup generated by [(1,2)(3,4), (1,4)(2,3)] of
 (Symmetric group of order 4! as a permutation group)

The socle of the socle is, essentially, the socle:

sage: s.socle() == s.subgroup(s.gens())
True

solvable_radical()#

Return the solvable radical of self.

The solvable radical (or just radical) of a group \(G\) is the largest solvable normal subgroup of \(G\).

EXAMPLES:

sage: G = SymmetricGroup(4)
sage: G.solvable_radical()
Subgroup generated by [(1,2), (1,2,3,4)] of
 (Symmetric group of order 4! as a permutation group)
sage: G = SymmetricGroup(5)
sage: G.solvable_radical()
Subgroup generated by [()] of
 (Symmetric group of order 5! as a permutation group)

stabilizer(point, action='OnPoints')#

Return the subgroup of self which stabilize the given position. self and its stabilizers must have same degree.

INPUT:

point – a point of the domain(), or a set of points depending on the value of action.
action – (string; default "OnPoints") should the group be considered to act on points (action="OnPoints") or on sets of points (action="OnSets")? In the latter case, the first argument must be a subset of domain().

EXAMPLES:

sage: G = PermutationGroup([ [(3,4)], [(1,3)] ])
sage: G.stabilizer(1)
Subgroup generated by [(3,4)] of (Permutation Group with generators [(3,4), (1,3)])
sage: G.stabilizer(3)
Subgroup generated by [(1,4)] of (Permutation Group with generators [(3,4), (1,3)])

The stabilizer of a set of points:

sage: s10 = groups.permutation.Symmetric(10)
sage: s10.stabilizer([1..3],"OnSets").cardinality()
30240
sage: factorial(3)*factorial(7)
30240

sage: G = PermutationGroup([[(1,2),(3,4)], [(1,2,3,4,10)]])
sage: G.stabilizer(10)
Subgroup generated by [(2,3,4), (1,2)(3,4)] of (Permutation Group with generators [(1,2)(3,4), (1,2,3,4,10)])
sage: G.stabilizer(1) == G.subgroup(['(2,3)(4,10)', '(2,10,3)'])
True
sage: G = PermutationGroup([[(2,3,4)],[(6,7)]])
sage: G.stabilizer(1)
Subgroup generated by [(6,7), (2,3,4)] of (Permutation Group with generators [(6,7), (2,3,4)])
sage: G.stabilizer(2)
Subgroup generated by [(6,7)] of (Permutation Group with generators [(6,7), (2,3,4)])
sage: G.stabilizer(3)
Subgroup generated by [(6,7)] of (Permutation Group with generators [(6,7), (2,3,4)])
sage: G.stabilizer(4)
Subgroup generated by [(6,7)] of (Permutation Group with generators [(6,7), (2,3,4)])
sage: G.stabilizer(5)
Subgroup generated by [(6,7), (2,3,4)] of (Permutation Group with generators [(6,7), (2,3,4)])
sage: G.stabilizer(6)
Subgroup generated by [(2,3,4)] of (Permutation Group with generators [(6,7), (2,3,4)])
sage: G.stabilizer(7)
Subgroup generated by [(2,3,4)] of (Permutation Group with generators [(6,7), (2,3,4)])
sage: G.stabilizer(8)
Traceback (most recent call last):
...
ValueError: 8 does not belong to the domain

sage: G = PermutationGroup([ [('c','d')], [('a','c')] ], domain='abcd')
sage: G.stabilizer('a')
Subgroup generated by [('c','d')] of (Permutation Group with generators [('c','d'), ('a','c')])
sage: G.stabilizer('b')
Subgroup generated by [('c','d'), ('a','c')] of (Permutation Group with generators [('c','d'), ('a','c')])
sage: G.stabilizer('c')
Subgroup generated by [('a','d')] of (Permutation Group with generators [('c','d'), ('a','c')])
sage: G.stabilizer('d')
Subgroup generated by [('a','c')] of (Permutation Group with generators [('c','d'), ('a','c')])

strong_generating_system(base_of_group=None, implementation='sage')#

Return a Strong Generating System of self according the given base for the right action of self on itself.

base_of_group is a list of the positions on which self acts, in any order. The algorithm returns a list of transversals and each transversal is a list of permutations. By default, base_of_group is [1, 2, 3, ..., d] where \(d\) is the degree of the group.

For base_of_group = \([ \mathrm{pos}_1, \mathrm{pos}_2, \dots , \mathrm{pos}_d]\) let \(G_i\) be the subgroup of \(G\) = self which stabilizes \(\mathrm{pos}_1, \mathrm{pos}_2, \dots , \mathrm{pos}_i\), so

\[G = G_0 \supset G_1 \supset G_2 \supset \dots \supset G_n = \{e\}\]

Then the algorithm returns \([ G_i.\mathrm{transversals}(\mathrm{pos}_{i+1})]_{1 \leq i \leq n}\)

INPUT:

base_of_group (optional) – (default: [1, 2, 3, ..., d]) a list containing the integers \(1, 2, \ldots , d\) in any order, where \(d\) is the degree of self
implementation – (default: "sage") either
- "sage" – use the direct implementation in Sage
- "gap" – if used, the base_of_group must be None and the computation is directly performed in GAP

OUTPUT:

A list of lists of permutations from the group, which form a strong generating system.

Warning

The outputs for implementations "sage" and "gap" differ: First, the output is reversed, and second, it might be that "sage" does not contain the trivial subgroup while "gap" does.

Also, both algorithms might yield different results based on the order in which base_of_group is given in the first situation.

EXAMPLES:

sage: G = PermutationGroup([[(7,8)],[(3,4)],[(4,5)]])
sage: G.strong_generating_system()
[[()], [()], [(), (3,4), (3,5,4)], [(), (4,5)], [()], [()], [(), (7,8)], [()]]
sage: G = PermutationGroup([[(1,2,3,4)],[(1,2)]])
sage: G.strong_generating_system()
[[(), (1,2)(3,4), (1,3)(2,4), (1,4)(2,3)],
[(), (2,4,3), (2,3,4)],
[(), (3,4)],
[()]]
sage: G = PermutationGroup([[(1,2,3)],[(4,5,7)],[(1,4,6)]])
sage: G.strong_generating_system()
[[(), (1,2,3), (1,4,6), (1,3,2), (1,5,7,4,6), (1,6,4), (1,7,5,4,6)],
 [(), (2,3,6), (2,6,3), (2,7,5,6,3), (2,5,6,3)(4,7), (2,4,5,6,3)],
 [(), (3,5,6), (3,4,7,5,6), (3,6)(5,7), (3,7,4,5,6)],
 [(), (4,7,5), (4,5,7), (4,6,7)],
 [(), (5,6,7), (5,7,6)], [()], [()]]
sage: G = PermutationGroup([[(1,2,3)],[(2,3,4)],[(3,4,5)]])
sage: G.strong_generating_system([5,4,3,2,1])
[[(), (1,5,3,4,2), (1,5,4,3,2), (1,5)(2,3), (1,5,2)],
 [(1,4)(2,3), (1,4,3), (1,4,2), ()],
 [(1,2,3), (1,3,2), ()], [()], [()]]
sage: G = PermutationGroup([[(3,4)]])
sage: G.strong_generating_system()
[[()], [()], [(), (3,4)], [()]]
sage: G.strong_generating_system(base_of_group=[3,1,2,4])
[[(), (3,4)], [()], [()], [()]]
sage: G = TransitiveGroup(12,17)
sage: G.strong_generating_system()
[[(), (1,4,11,2)(3,6,5,8)(7,10,9,12), (1,8,3,2)(4,11,10,9)(5,12,7,6),
  (1,7)(2,8)(3,9)(4,10)(5,11)(6,12), (1,12,7,2)(3,10,9,8)(4,11,6,5),
  (1,11)(2,8)(3,5)(4,10)(6,12)(7,9), (1,10,11,8)(2,3,12,5)(4,9,6,7),
  (1,3)(2,8)(4,10)(5,7)(6,12)(9,11), (1,2,3,8)(4,9,10,11)(5,6,7,12),
  (1,6,7,8)(2,3,4,9)(5,10,11,12), (1,5,9)(3,11,7), (1,9,5)(3,7,11)],
 [(), (2,6,10)(4,12,8), (2,10,6)(4,8,12)],
 [()], [()], [()], [()], [()], [()], [()], [()], [()], [()]]

sage: A = PermutationGroup([(1,2),(1,2,3,4,5,6,7,8,9)])
sage: X = A.strong_generating_system()
sage: Y = A.strong_generating_system(implementation="gap")
sage: [len(x) for x in X]
[9, 8, 7, 6, 5, 4, 3, 2, 1]
sage: [len(y) for y in Y]
[1, 2, 3, 4, 5, 6, 7, 8, 9]

structure_description(G, latex=False)#

Return a string that tries to describe the structure of G.

This methods wraps GAP’s StructureDescription method.

For full details, including the form of the returned string and the algorithm to build it, see GAP’s documentation.

INPUT:

latex – a boolean (default: False). If True, return a LaTeX formatted string.

OUTPUT:

string

Warning

From GAP’s documentation: The string returned by StructureDescription is not an isomorphism invariant: non-isomorphic groups can have the same string value, and two isomorphic groups in different representations can produce different strings.

EXAMPLES:

sage: # needs sage.groups
sage: G = CyclicPermutationGroup(6)
sage: G.structure_description()
'C6'
sage: G.structure_description(latex=True)
'C_{6}'
sage: G2 = G.direct_product(G, maps=False)
sage: LatexExpr(G2.structure_description(latex=True))
C_{6} \times C_{6}

This method is mainly intended for small groups or groups with few normal subgroups. Even then there are some surprises:

sage: D3 = DihedralGroup(3)                                                     # needs sage.groups
sage: D3.structure_description()                                                # needs sage.groups
'S3'

We use the Sage notation for the degree of dihedral groups:

sage: D4 = DihedralGroup(4)                                                     # needs sage.groups
sage: D4.structure_description()                                                # needs sage.groups
'D4'

Works for finitely presented groups (github issue #17573):

sage: F.<x, y> = FreeGroup()                                                    # needs sage.groups
sage: G = F / [x^2*y^-1, x^3*y^2, x*y*x^-1*y^-1]                                # needs sage.groups
sage: G.structure_description()                                                 # needs sage.groups
'C7'

And matrix groups (github issue #17573):

sage: groups.matrix.GL(4,2).structure_description()                             # needs sage.libs.gap sage.modules
'A8'

subgroup(gens=None, gap_group=None, domain=None, category=None, canonicalize=True, check=True)#

Wraps the PermutationGroup_subgroup constructor. The argument gens is a list of elements of self.

EXAMPLES:

sage: G = PermutationGroup([(1,2,3),(3,4,5)])
sage: g = G((1,2,3))
sage: G.subgroup([g])
Subgroup generated by [(1,2,3)] of
 (Permutation Group with generators [(3,4,5), (1,2,3)])

subgroups()#

Return a list of all the subgroups of self.

OUTPUT:

Each possible subgroup of self is contained once in the returned list. The list is in order, according to the size of the subgroups, from the trivial subgroup with one element on through up to the whole group. Conjugacy classes of subgroups are contiguous in the list.

Warning

For even relatively small groups this method can take a very long time to execute, or create vast amounts of output. Likely both. Its purpose is instructional, as it can be useful for studying small groups. The 156 subgroups of the full symmetric group on 5 symbols of order 120, \(S_5\), can be computed in about a minute on commodity hardware in 2011. The 64 subgroups of the cyclic group of order \(30030 = 2\cdot 3\cdot 5\cdot 7\cdot 11\cdot 13\) takes about twice as long.

For faster results, which still exhibit the structure of the possible subgroups, use conjugacy_classes_subgroups().

EXAMPLES:

sage: G = SymmetricGroup(3)
sage: G.subgroups()
[Subgroup generated by [()] of
  (Symmetric group of order 3! as a permutation group),
 Subgroup generated by [(2,3)] of
  (Symmetric group of order 3! as a permutation group),
 Subgroup generated by [(1,2)] of
  (Symmetric group of order 3! as a permutation group),
 Subgroup generated by [(1,3)] of
  (Symmetric group of order 3! as a permutation group),
 Subgroup generated by [(1,2,3)] of
  (Symmetric group of order 3! as a permutation group),
 Subgroup generated by [(2,3), (1,2,3)] of
  (Symmetric group of order 3! as a permutation group)]

sage: G = CyclicPermutationGroup(14)
sage: G.subgroups()
[Subgroup generated by [()] of
  (Cyclic group of order 14 as a permutation group),
 Subgroup generated by [(1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)] of
  (Cyclic group of order 14 as a permutation group),
 Subgroup generated by [(1,3,5,7,9,11,13)(2,4,6,8,10,12,14)] of
  (Cyclic group of order 14 as a permutation group),
 Subgroup generated by [(1,2,3,4,5,6,7,8,9,10,11,12,13,14),
                        (1,3,5,7,9,11,13)(2,4,6,8,10,12,14)] of
  (Cyclic group of order 14 as a permutation group)]

AUTHOR:

Rob Beezer (2011-01-24)

sylow_subgroup(p)#

Return a Sylow \(p\)-subgroup of the finite group \(G\), where \(p\) is a prime.

This is a \(p\)-subgroup of \(G\) whose index in \(G\) is coprime to \(p\).

Wraps the GAP function SylowSubgroup.

EXAMPLES:

sage: G = PermutationGroup(['(1,2,3)', '(2,3)'])
sage: G.sylow_subgroup(2)
Subgroup generated by [(2,3)] of
 (Permutation Group with generators [(2,3), (1,2,3)])
sage: G.sylow_subgroup(5)
Subgroup generated by [()] of
 (Permutation Group with generators [(2,3), (1,2,3)])

transversals(point)#

If G is a permutation group acting on the set \(X = \{1, 2, ...., n\}\) and H is the stabilizer subgroup of <integer>, a right (respectively left) transversal is a set containing exactly one element from each right (respectively left) coset of H. This method returns a right transversal of self by the stabilizer of self on <integer> position.

EXAMPLES:

sage: G = PermutationGroup([ [(3,4)], [(1,3)] ])
sage: G.transversals(1)
[(), (1,3,4), (1,4,3)]
sage: G = PermutationGroup([[(1,2),(3,4)], [(1,2,3,4,10)]])
sage: G.transversals(1)
[(), (1,2)(3,4), (1,3,2,10,4), (1,4,2,10,3), (1,10,4,3,2)]

sage: G = PermutationGroup([ [('c','d')], [('a','c')] ])
sage: G.transversals('a')
[(), ('a','c','d'), ('a','d','c')]

trivial_character()#

Return the trivial character of self.

EXAMPLES:

sage: SymmetricGroup(3).trivial_character()                                 # needs sage.rings.number_field
Character of Symmetric group of order 3! as a permutation group

upper_central_series()#

Return the upper central series of this group as a list of permutation groups.

EXAMPLES:

These computations use pseudo-random numbers, so we set the seed for reproducible testing:

sage: G = PermutationGroup([[(1,2,3),(4,5)],[(3,4)]])
sage: G.upper_central_series()
[Subgroup generated by [()] of
 (Permutation Group with generators [(3,4), (1,2,3)(4,5)])]

class sage.groups.perm_gps.permgroup.PermutationGroup_subgroup(ambient, gens=None, gap_group=None, domain=None, category=None, canonicalize=True, check=True)#

Bases: PermutationGroup_generic

Subgroup subclass of PermutationGroup_generic, so instance methods are inherited.

EXAMPLES:

sage: G = CyclicPermutationGroup(4)
sage: gens = G.gens()
sage: H = DihedralGroup(4)
sage: H.subgroup(gens)
Subgroup generated by [(1,2,3,4)] of
 (Dihedral group of order 8 as a permutation group)
sage: K = H.subgroup(gens)
sage: K.list()
[(), (1,2,3,4), (1,3)(2,4), (1,4,3,2)]
sage: K.ambient_group()
Dihedral group of order 8 as a permutation group
sage: K.gens()
((1,2,3,4),)

ambient_group()#

Return the ambient group related to self.

EXAMPLES:

An example involving the dihedral group on four elements, \(D_8\):

sage: G = DihedralGroup(4)
sage: H = CyclicPermutationGroup(4)
sage: gens = H.gens()
sage: S = PermutationGroup_subgroup(G, list(gens))
sage: S.ambient_group()
Dihedral group of order 8 as a permutation group
sage: S.ambient_group() == G
True

is_normal(other=None)#

Return True if this group is a normal subgroup of other. If other is not specified, then it is assumed to be the ambient group.

EXAMPLES:

sage: S = SymmetricGroup(['a','b','c'])
sage: H = S.subgroup([('a', 'b', 'c')]); H
Subgroup generated by [('a','b','c')] of
 (Symmetric group of order 3! as a permutation group)
sage: H.is_normal()
True

sage.groups.perm_gps.permgroup.direct_product_permgroups(P)#

Takes the direct product of the permutation groups listed in P.

EXAMPLES:

sage: G1 = AlternatingGroup([1,2,4,5])
sage: G2 = AlternatingGroup([3,4,6,7])
sage: D = direct_product_permgroups([G1,G2,G1])
sage: D.order()
1728
sage: D = direct_product_permgroups([G1])
sage: D == G1
True
sage: direct_product_permgroups([])
Symmetric group of order 0! as a permutation group

sage.groups.perm_gps.permgroup.from_gap_list(G, src)#

Convert a string giving a list of GAP permutations into a list of elements of G.

EXAMPLES:

sage: from sage.groups.perm_gps.permgroup import from_gap_list
sage: G = PermutationGroup([[(1,2,3),(4,5)],[(3,4)]])
sage: L = from_gap_list(G, "[(1,2,3)(4,5), (3,4)]"); L
[(1,2,3)(4,5), (3,4)]
sage: L[0].parent() is G
True
sage: L[1].parent() is G
True

sage.groups.perm_gps.permgroup.hap_decorator(f)#

A decorator for permutation group methods that require HAP. It checks to see that HAP is installed as well as checks that the argument p is either 0 or prime.

EXAMPLES:

sage: # optional - gap_package_hap
sage: from sage.groups.perm_gps.permgroup import hap_decorator
sage: def foo(self, n, p=0): print("Done")
sage: foo = hap_decorator(foo)
sage: foo(None, 3)
Done
sage: foo(None, 3, 0)
Done
sage: foo(None, 3, 5)
Done
sage: foo(None, 3, 4)
Traceback (most recent call last):
...
ValueError: p must be 0 or prime

sage.groups.perm_gps.permgroup.load_hap()#

Load the GAP hap package into the default GAP interpreter interface.

EXAMPLES:

sage: sage.groups.perm_gps.permgroup.load_hap()         # optional - gap_package_hap