Groups#

class sage.categories.groups.Groups(base_category)#

Bases: CategoryWithAxiom_singleton

The category of (multiplicative) groups, i.e. monoids with inverses.

EXAMPLES:

sage: Groups()
Category of groups
sage: Groups().super_categories()
[Category of monoids, Category of inverse unital magmas]
Algebras#

alias of GroupAlgebras

class CartesianProducts(category, *args)#

Bases: CartesianProductsCategory

The category of groups constructed as Cartesian products of groups.

This construction gives the direct product of groups. See Wikipedia article Direct_product and Wikipedia article Direct_product_of_groups for more information.

class ParentMethods#

Bases: object

group_generators()#

Return the group generators of self.

EXAMPLES:

sage: # needs sage.groups
sage: C5 = CyclicPermutationGroup(5)
sage: C4 = CyclicPermutationGroup(4)
sage: S4 = SymmetricGroup(3)
sage: C = cartesian_product([C5, C4, S4])
sage: C.group_generators()
Family (((1,2,3,4,5), (), ()),
        ((), (1,2,3,4), ()),
        ((), (), (1,2)),
        ((), (), (2,3)))

We check the other portion of github issue #16718 is fixed:

sage: len(C.j_classes())                                            # needs sage.groups
1

An example with an infinitely generated group (a better output is needed):

sage: # needs sage.groups
sage: G = Groups.free([1,2])
sage: H = Groups.free(ZZ)
sage: C = cartesian_product([G, H])
sage: C.monoid_generators()
Lazy family (gen(i))_{i in The Cartesian product of (...)}
order()#

Return the cardinality of self.

EXAMPLES:

sage: C = cartesian_product([SymmetricGroup(10), SL(2, GF(3))])     # needs sage.groups sage.rings.finite_rings
sage: C.order()                                                     # needs sage.groups sage.rings.finite_rings
87091200

Todo

this method is just here to prevent FiniteGroups.ParentMethods to call _cardinality_from_iterator.

extra_super_categories()#

A Cartesian product of groups is endowed with a natural group structure.

EXAMPLES:

sage: C = Groups().CartesianProducts()
sage: C.extra_super_categories()
[Category of groups]
sage: sorted(C.super_categories(), key=str)
[Category of Cartesian products of inverse unital magmas,
 Category of Cartesian products of monoids,
 Category of groups]
class Commutative(base_category)#

Bases: CategoryWithAxiom

Category of commutative (abelian) groups.

A group \(G\) is commutative if \(xy = yx\) for all \(x,y \in G\).

static free(index_set=None, names=None, **kwds)#

Return the free commutative group.

INPUT:

  • index_set – (optional) an index set for the generators; if an integer, then this represents \(\{0, 1, \ldots, n-1\}\)

  • names – a string or list/tuple/iterable of strings (default: 'x'); the generator names or name prefix

EXAMPLES:

sage: # needs sage.groups
sage: Groups.Commutative.free(index_set=ZZ)
Free abelian group indexed by Integer Ring
sage: Groups().Commutative().free(ZZ)
Free abelian group indexed by Integer Ring
sage: Groups().Commutative().free(5)
Multiplicative Abelian group isomorphic to Z x Z x Z x Z x Z
sage: F.<x,y,z> = Groups().Commutative().free(); F
Multiplicative Abelian group isomorphic to Z x Z x Z
class ElementMethods#

Bases: object

conjugacy_class()#

Return the conjugacy class of self.

EXAMPLES:

sage: D = DihedralGroup(5)                                              # needs sage.groups
sage: g = D((1,3,5,2,4))                                                # needs sage.groups
sage: g.conjugacy_class()                                               # needs sage.groups
Conjugacy class of (1,3,5,2,4)
 in Dihedral group of order 10 as a permutation group

sage: H = MatrixGroup([matrix(GF(5), 2, [1,2, -1,1]),                   # needs sage.modules
....:                  matrix(GF(5), 2, [1,1, 0,1])])
sage: h = H(matrix(GF(5), 2, [1,2, -1,1]))                              # needs sage.modules
sage: h.conjugacy_class()                                               # needs sage.groups sage.modules
Conjugacy class of [1 2]
                   [4 1]
in Matrix group over Finite Field of size 5 with 2 generators (
[1 2]  [1 1]
[4 1], [0 1]
)

sage: G = SL(2, GF(2))                                                  # needs sage.modules
sage: g = G.gens()[0]                                                   # needs sage.modules
sage: g.conjugacy_class()                                               # needs sage.modules
Conjugacy class of [1 1]
[0 1] in Special Linear Group of degree 2 over Finite Field of size 2

sage: G = SL(2, QQ)                                                     # needs sage.modules
sage: g = G([[1,1], [0,1]])                                             # needs sage.modules
sage: g.conjugacy_class()                                               # needs sage.groups sage.modules
Conjugacy class of [1 1]
[0 1] in Special Linear Group of degree 2 over Rational Field
Finite#

alias of FiniteGroups

Lie#

alias of LieGroups

class ParentMethods#

Bases: object

cayley_table(names='letters', elements=None)#

Return the “multiplication” table of this multiplicative group, which is also known as the “Cayley table”.

Note

The order of the elements in the row and column headings is equal to the order given by the table’s column_keys() method. The association between the actual elements and the names/symbols used in the table can also be retrieved as a dictionary with the translation() method.

For groups, this routine should behave identically to the multiplication_table() method for magmas, which applies in greater generality.

INPUT:

  • names - the type of names used, values are:

    • 'letters' - lowercase ASCII letters are used for a base 26 representation of the elements’ positions in the list given by list(), padded to a common width with leading ‘a’s.

    • 'digits' - base 10 representation of the elements’ positions in the list given by column_keys(), padded to a common width with leading zeros.

    • 'elements' - the string representations of the elements themselves.

    • a list - a list of strings, where the length of the list equals the number of elements.

  • elements - default = None. A list of elements of the group, in forms that can be coerced into the structure, eg. their string representations. This may be used to impose an alternate ordering on the elements, perhaps when this is used in the context of a particular structure. The default is to use whatever ordering is provided by the the group, which is reported by the column_keys() method. Or the elements can be a subset which is closed under the operation. In particular, this can be used when the base set is infinite.

OUTPUT:

An object representing the multiplication table. This is an OperationTable object and even more documentation can be found there.

EXAMPLES:

Permutation groups, matrix groups and abelian groups can all compute their multiplication tables.

sage: # needs sage.groups
sage: G = DiCyclicGroup(3)
sage: T = G.cayley_table()
sage: T.column_keys()
((), (5,6,7), ..., (1,4,2,3)(5,7))
sage: T
*  a b c d e f g h i j k l
 +------------------------
a| a b c d e f g h i j k l
b| b c a e f d i g h l j k
c| c a b f d e h i g k l j
d| d e f a b c j k l g h i
e| e f d b c a l j k i g h
f| f d e c a b k l j h i g
g| g h i j k l d e f a b c
h| h i g k l j f d e c a b
i| i g h l j k e f d b c a
j| j k l g h i a b c d e f
k| k l j h i g c a b f d e
l| l j k i g h b c a e f d

sage: M = SL(2, 2)                                                      # needs sage.modules
sage: M.cayley_table()                                                  # needs sage.modules
*  a b c d e f
 +------------
a| a b c d e f
b| b a d c f e
c| c e a f b d
d| d f b e a c
e| e c f a d b
f| f d e b c a

sage: A = AbelianGroup([2, 3])                                          # needs sage.groups
sage: A.cayley_table()                                                  # needs sage.groups
*  a b c d e f
 +------------
a| a b c d e f
b| b c a e f d
c| c a b f d e
d| d e f a b c
e| e f d b c a
f| f d e c a b

Lowercase ASCII letters are the default symbols used for the table, but you can also specify the use of decimal digit strings, or provide your own strings (in the proper order if they have meaning). Also, if the elements themselves are not too complex, you can choose to just use the string representations of the elements themselves.

sage: C = CyclicPermutationGroup(11)                                    # needs sage.groups
sage: C.cayley_table(names='digits')                                    # needs sage.groups
 *  00 01 02 03 04 05 06 07 08 09 10
  +---------------------------------
00| 00 01 02 03 04 05 06 07 08 09 10
01| 01 02 03 04 05 06 07 08 09 10 00
02| 02 03 04 05 06 07 08 09 10 00 01
03| 03 04 05 06 07 08 09 10 00 01 02
04| 04 05 06 07 08 09 10 00 01 02 03
05| 05 06 07 08 09 10 00 01 02 03 04
06| 06 07 08 09 10 00 01 02 03 04 05
07| 07 08 09 10 00 01 02 03 04 05 06
08| 08 09 10 00 01 02 03 04 05 06 07
09| 09 10 00 01 02 03 04 05 06 07 08
10| 10 00 01 02 03 04 05 06 07 08 09

sage: G = QuaternionGroup()                                             # needs sage.groups
sage: names = ['1', 'I', '-1', '-I', 'J', '-K', '-J', 'K']
sage: G.cayley_table(names=names)                                       # needs sage.groups
 *   1  I -1 -I  J -K -J  K
  +------------------------
 1|  1  I -1 -I  J -K -J  K
 I|  I -1 -I  1  K  J -K -J
-1| -1 -I  1  I -J  K  J -K
-I| -I  1  I -1 -K -J  K  J
 J|  J -K -J  K -1 -I  1  I
-K| -K -J  K  J  I -1 -I  1
-J| -J  K  J -K  1  I -1 -I
 K|  K  J -K -J -I  1  I -1

sage: A = AbelianGroup([2, 2])                                          # needs sage.groups
sage: A.cayley_table(names='elements')                                  # needs sage.groups
    *      1    f1    f0 f0*f1
     +------------------------
    1|     1    f1    f0 f0*f1
   f1|    f1     1 f0*f1    f0
   f0|    f0 f0*f1     1    f1
f0*f1| f0*f1    f0    f1     1

The change_names() routine behaves similarly, but changes an existing table “in-place.”

sage: # needs sage.groups
sage: G = AlternatingGroup(3)
sage: T = G.cayley_table()
sage: T.change_names('digits')
sage: T
*  0 1 2
 +------
0| 0 1 2
1| 1 2 0
2| 2 0 1

For an infinite group, you can still work with finite sets of elements, provided the set is closed under multiplication. Elements will be coerced into the group as part of setting up the table.

sage: # needs sage.modules
sage: G = SL(2,ZZ); G
Special Linear Group of degree 2 over Integer Ring
sage: identity = matrix(ZZ, [[1,0], [0,1]])
sage: G.cayley_table(elements=[identity, -identity])
*  a b
 +----
a| a b
b| b a

The OperationTable class provides even greater flexibility, including changing the operation. Here is one such example, illustrating the computation of commutators. commutator is defined as a function of two variables, before being used to build the table. From this, the commutator subgroup seems obvious, and creating a Cayley table with just these three elements confirms that they form a closed subset in the group.

sage: # needs sage.groups sage.modules
sage: from sage.matrix.operation_table import OperationTable
sage: G = DiCyclicGroup(3)
sage: commutator = lambda x, y: x*y*x^-1*y^-1
sage: T = OperationTable(G, commutator); T
.  a b c d e f g h i j k l
 +------------------------
a| a a a a a a a a a a a a
b| a a a a a a c c c c c c
c| a a a a a a b b b b b b
d| a a a a a a a a a a a a
e| a a a a a a c c c c c c
f| a a a a a a b b b b b b
g| a b c a b c a c b a c b
h| a b c a b c b a c b a c
i| a b c a b c c b a c b a
j| a b c a b c a c b a c b
k| a b c a b c b a c b a c
l| a b c a b c c b a c b a
sage: trans = T.translation()
sage: comm = [trans['a'], trans['b'], trans['c']]
sage: comm
[(), (5,6,7), (5,7,6)]
sage: P = G.cayley_table(elements=comm)
sage: P
*  a b c
 +------
a| a b c
b| b c a
c| c a b

Todo

Arrange an ordering of elements into cosets of a normal subgroup close to size \(\sqrt{n}\). Then the quotient group structure is often apparent in the table. See comments on github issue #7555.

AUTHOR:

  • Rob Beezer (2010-03-15)

conjugacy_class(g)#

Return the conjugacy class of the element g.

This is a fall-back method for groups not defined over GAP.

EXAMPLES:

sage: A = AbelianGroup([2, 2])                                          # needs sage.groups
sage: c = A.conjugacy_class(A.an_element())                             # needs sage.groups
sage: type(c)                                                           # needs sage.groups
<class 'sage.groups.conjugacy_classes.ConjugacyClass_with_category'>
group_generators()#

Return group generators for self.

This default implementation calls gens(), for backward compatibility.

EXAMPLES:

sage: A = AlternatingGroup(4)                                           # needs sage.groups
sage: A.group_generators()                                              # needs sage.groups
Family ((2,3,4), (1,2,3))
holomorph()#

The holomorph of a 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)

EXAMPLES:

sage: G = Groups().example()                                            # needs sage.modules
sage: G.holomorph()                                                     # needs sage.modules
Traceback (most recent call last):
...
NotImplementedError: holomorph of General Linear Group of degree 4
over Rational Field not yet implemented
monoid_generators()#

Return the generators of self as a monoid.

Let \(G\) be a group with generating set \(X\). In general, the generating set of \(G\) as a monoid is given by \(X \cup X^{-1}\), where \(X^{-1}\) is the set of inverses of \(X\). If \(G\) is a finite group, then the generating set as a monoid is \(X\).

EXAMPLES:

sage: # needs sage.groups
sage: A = AlternatingGroup(4)
sage: A.monoid_generators()
Family ((2,3,4), (1,2,3))
sage: F.<x,y> = FreeGroup()
sage: F.monoid_generators()
Family (x, y, x^-1, y^-1)
semidirect_product(N, mapping, check=True)#

The semi-direct product of two groups

EXAMPLES:

sage: G = Groups().example()                                            # needs sage.modules
sage: G.semidirect_product(G, Morphism(G, G))                           # needs sage.modules
Traceback (most recent call last):
...
NotImplementedError: semidirect product of General Linear Group of degree 4
over Rational Field and General Linear Group of degree 4 over Rational Field
not yet implemented
class Topological(category, *args)#

Bases: TopologicalSpacesCategory

Category of topological groups.

A topological group \(G\) is a group which has a topology such that multiplication and taking inverses are continuous functions.

REFERENCES:

example()#

EXAMPLES:

sage: Groups().example()                                                    # needs sage.modules
General Linear Group of degree 4 over Rational Field
static free(index_set=None, names=None, **kwds)#

Return the free group.

INPUT:

  • index_set – (optional) an index set for the generators; if an integer, then this represents \(\{0, 1, \ldots, n-1\}\)

  • names – a string or list/tuple/iterable of strings (default: 'x'); the generator names or name prefix

When the index set is an integer or only variable names are given, this returns FreeGroup_class, which currently has more features due to the interface with GAP than IndexedFreeGroup.

EXAMPLES:

sage: # needs sage.groups
sage: Groups.free(index_set=ZZ)
Free group indexed by Integer Ring
sage: Groups().free(ZZ)
Free group indexed by Integer Ring
sage: Groups().free(5)
Free Group on generators {x0, x1, x2, x3, x4}
sage: F.<x,y,z> = Groups().free(); F
Free Group on generators {x, y, z}