Multiplicative Abelian Groups#

This module lets you compute with finitely generated Abelian groups of the form

\[G = \ZZ^r \oplus \ZZ_{k_1} \oplus \cdots \oplus \ZZ_{k_t}\]

It is customary to denote the infinite cyclic group \(\ZZ\) as having order \(0\), so the data defining the Abelian group can be written as an integer vector

\[\vec{k} = (0, \dots, 0, k_1, \dots, k_t)\]

where there are \(r\) zeroes and \(t\) non-zero values. To construct this Abelian group in Sage, you can either specify all entries of \(\vec{k}\) or only the non-zero entries together with the total number of generators:

sage: AbelianGroup([0,0,0,2,3])
Multiplicative Abelian group isomorphic to Z x Z x Z x C2 x C3
sage: AbelianGroup(5, [2,3])
Multiplicative Abelian group isomorphic to Z x Z x Z x C2 x C3

It is also legal to specify \(1\) as the order. The corresponding generator will be the neutral element, but it will still take up an index in the labelling of the generators:

sage: G = AbelianGroup([2,1,3], names='g')
sage: G.gens()
(g0, 1, g2)

Note that this presentation is not unique, for example \(\ZZ_6 \cong \ZZ_2 \times \ZZ_3\). The orders of the generators \(\vec{k}=(0,\dots,0,k_1,\dots, k_t)\) has previously been called invariants in Sage, even though they are not necessarily the (unique) invariant factors of the group. You should now use gens_orders() instead:

sage: J = AbelianGroup([2,0,3,2,4]);  J
Multiplicative Abelian group isomorphic to C2 x Z x C3 x C2 x C4
sage: J.gens_orders()            # use this instead
(2, 0, 3, 2, 4)
sage: J.invariants()             # deprecated
(2, 0, 3, 2, 4)
sage: J.elementary_divisors()    # these are the "invariant factors"
(2, 2, 12, 0)
sage: for i in range(J.ngens()):
....:     print((i, J.gen(i), J.gen(i).order()))     # or use this form
(0, f0, 2)
(1, f1, +Infinity)
(2, f2, 3)
(3, f3, 2)
(4, f4, 4)

Background on invariant factors and the Smith normal form (according to section 4.1 of [Cohen1]): An abelian group is a group \(A\) for which there exists an exact sequence \(\ZZ^k \rightarrow \ZZ^\ell \rightarrow A \rightarrow 1\), for some positive integers \(k,\ell\) with \(k\leq \ell\). For example, a finite abelian group has a decomposition

\[A = \langle a_1\rangle \times \dots \times \langle a_\ell\rangle ,\]

where \(\mathrm{ord}(a_i)=p_i^{c_i}\), for some primes \(p_i\) and some positive integers \(c_i\), \(i=1,...,\ell\). GAP calls the list (ordered by size) of the \(p_i^{c_i}\) the abelian invariants. In Sage they will be called invariants. In this situation, \(k=\ell\) and \(\phi: \ZZ^\ell \rightarrow A\) is the map \(\phi(x_1,...,x_\ell) = a_1^{x_1}...a_\ell^{x_\ell}\), for \((x_1,...,x_\ell)\in \ZZ^\ell\). The matrix of relations \(M:\ZZ^k \rightarrow \ZZ^\ell\) is the matrix whose rows generate the kernel of \(\phi\) as a \(\ZZ\)-module. In other words, \(M=(M_{ij})\) is a \(\ell\times \ell\) diagonal matrix with \(M_{ii}=p_i^{c_i}\). Consider now the subgroup \(B\subset A\) generated by \(b_1 = a_1^{f_{1,1}}...a_\ell^{f_{\ell,1}}\), …, \(b_m = a_1^{f_{1,m}}...a_\ell^{f_{\ell,m}}\). The kernel of the map \(\phi_B: \ZZ^m \rightarrow B\) defined by \(\phi_B(y_1,...,y_m) = b_1^{y_1}...b_m^{y_m}\), for \((y_1,...,y_m)\in \ZZ^m\), is the kernel of the matrix

\[\begin{split}F= \left( \begin{array}{cccc} f_{11} & f_{12} & \dots & f_{1m}\\ f_{21} & f_{22} & \dots & f_{2m}\\ \vdots & & \ddots & \vdots \\ f_{\ell,1} & f_{\ell,2} & \dots & f_{\ell,m} \end{array} \right),\end{split}\]

regarded as a map \(\ZZ^m\rightarrow (\ZZ/p_1^{c_1}\ZZ)\times ...\times (\ZZ/p_\ell^{c_\ell}\ZZ)\). In particular, \(B\cong \ZZ^m/\ker(F)\). If \(B=A\) then the Smith normal form (SNF) of a generator matrix of \(\ker(F)\) and the SNF of \(M\) are the same. The diagonal entries \(s_i\) of the SNF \(S = \mathrm{diag}[s_1,s_2,s_3, ... s_r,0,0,...0]\), are called determinantal divisors of \(F\). where \(r\) is the rank. The invariant factors of \(A\) are:

\[s_1, s_2/s_1, s_3/s_2, ... s_r/s_{r-1}.\]

Sage supports multiplicative abelian groups on any prescribed finite number \(n \geq 0\) of generators. Use the AbelianGroup() function to create an abelian group, and the gen() and gens() methods to obtain the corresponding generators. You can print the generators as arbitrary strings using the optional names argument to the AbelianGroup() function.

EXAMPLE 1:

We create an abelian group in zero or more variables; the syntax T(1) creates the identity element even in the rank zero case:

sage: T = AbelianGroup(0, [])
sage: T
Trivial Abelian group
sage: T.gens()
()
sage: T(1)
1

EXAMPLE 2:

An Abelian group uses a multiplicative representation of elements, but the underlying representation is lists of integer exponents:

sage: F = AbelianGroup(5, [3,4,5,5,7], names = list("abcde"))
sage: F
Multiplicative Abelian group isomorphic to C3 x C4 x C5 x C5 x C7
sage: (a,b,c,d,e) = F.gens()
sage: a*b^2*e*d
a*b^2*d*e
sage: x = b^2*e*d*a^7
sage: x
a*b^2*d*e
sage: x.list()
[1, 2, 0, 1, 1]

REFERENCES:

[Cohen1]

H. Cohen, Advanced topics in computational number theory, Springer, 2000.

[Cohen2]

H. Cohen, A course in computational algebraic number theory, Springer, 1996.

[Rotman]

J. Rotman, An introduction to the theory of groups, 4th ed, Springer, 1995.

Warning

Many basic properties for infinite abelian groups are not implemented.

AUTHORS:

William Stein, David Joyner (2008-12): added (user requested) is_cyclic, fixed elementary_divisors.
David Joyner (2006-03): (based on free abelian monoids by David Kohel)
David Joyner (2006-05) several significant bug fixes
David Joyner (2006-08) trivial changes to docs, added random, fixed bug in how invariants are recorded
David Joyner (2006-10) added dual_group method
David Joyner (2008-02) fixed serious bug in word_problem
David Joyner (2008-03) fixed bug in trivial group case
David Loeffler (2009-05) added subgroups method
Volker Braun (2012-11) port to new Parent base. Use tuples for immutables. Rename invariants to gens_orders.

sage.groups.abelian_gps.abelian_group.AbelianGroup(n, gens_orders=None, names='f')#

Create the multiplicative abelian group in \(n\) generators with given orders of generators (which need not be prime powers).

INPUT:

n – integer (optional). If not specified, will be derived
from gens_orders.
gens_orders – a list of non-negative integers in the form
\([a_0, a_1, \dots, a_{n-1}]\), typically written in increasing order. This list is padded with zeros if it has length less than \(n\). The orders of the commuting generators, with \(0\) denoting an infinite cyclic factor.
names – (optional) names of generators

Alternatively, you can also give input in the form AbelianGroup(gens_orders, names="f"), where the names keyword argument must be explicitly named.

OUTPUT:

Abelian group with generators and invariant type. The default name for generator A.i is fi, as in GAP.

EXAMPLES:

sage: F = AbelianGroup(5, [5,5,7,8,9], names='abcde')
sage: F(1)
1
sage: (a, b, c, d, e) = F.gens()
sage: mul([ a, b, a, c, b, d, c, d ], F(1))
a^2*b^2*c^2*d^2
sage: d * b**2 * c**3
b^2*c^3*d
sage: F = AbelianGroup(3, [2]*3); F
Multiplicative Abelian group isomorphic to C2 x C2 x C2
sage: H = AbelianGroup([2,3], names="xy"); H
Multiplicative Abelian group isomorphic to C2 x C3
sage: AbelianGroup(5)
Multiplicative Abelian group isomorphic to Z x Z x Z x Z x Z
sage: AbelianGroup(5).order()
+Infinity

Notice that \(0\)’s are prepended if necessary:

sage: G = AbelianGroup(5, [2,3,4]);  G
Multiplicative Abelian group isomorphic to Z x Z x C2 x C3 x C4
sage: G.gens_orders()
(0, 0, 2, 3, 4)

The invariant list must not be longer than the number of generators:

sage: AbelianGroup(2, [2,3,4])
Traceback (most recent call last):
...
ValueError: gens_orders (=(2, 3, 4)) must have length n (=2)

class sage.groups.abelian_gps.abelian_group.AbelianGroup_class(generator_orders, names, category=None)#

Bases: UniqueRepresentation, AbelianGroup

The parent for Abelian groups with chosen generator orders.

Warning

You should use AbelianGroup() to construct Abelian groups and not instantiate this class directly.

INPUT:

generator_orders – list of integers. The orders of the (commuting) generators. Zero denotes an infinite cyclic generator.
names – names of the group generators (optional).

EXAMPLES:

sage: Z2xZ3 = AbelianGroup([2,3])
sage: Z6 = AbelianGroup([6])
sage: Z2xZ3 is Z2xZ3, Z6 is Z6
(True, True)
sage: Z2xZ3 is Z6
False
sage: Z2xZ3 == Z6
False
sage: Z2xZ3.is_isomorphic(Z6)
True

sage: F = AbelianGroup(5,[5,5,7,8,9], names=list("abcde")); F
Multiplicative Abelian group isomorphic to C5 x C5 x C7 x C8 x C9
sage: F = AbelianGroup(5,[2, 4, 12, 24, 120], names=list("abcde")); F
Multiplicative Abelian group isomorphic to C2 x C4 x C12 x C24 x C120
sage: F.elementary_divisors()
(2, 4, 12, 24, 120)

sage: F.category()
Category of finite enumerated commutative groups

Element#: alias of AbelianGroupElement

Subgroup#: alias of AbelianGroup_subgroup

cardinality()#

Return the order of this group.

EXAMPLES:

sage: G = AbelianGroup(2, [2,3])
sage: G.order()
6
sage: G = AbelianGroup(3, [2,3,0])
sage: G.order()
+Infinity

dual_group(names='X', base_ring=None)#

Return the dual group.

INPUT:

names – string or list of strings. The generator names for the dual group.
base_ring – the base ring. If None (default), then a suitable cyclotomic field is picked automatically.

OUTPUT:

The dual abelian group.

EXAMPLES:

sage: G = AbelianGroup([2])
sage: G.dual_group()                                                        # needs sage.rings.number_field
Dual of Abelian Group isomorphic to Z/2Z over Cyclotomic Field of order 2 and degree 1
sage: G.dual_group().gens()                                                 # needs sage.rings.number_field
(X,)
sage: G.dual_group(names='Z').gens()                                        # needs sage.rings.number_field
(Z,)

sage: G.dual_group(base_ring=QQ)
Dual of Abelian Group isomorphic to Z/2Z over Rational Field

elementary_divisors()#

This returns the elementary divisors of the group, using Pari.

Note

Here is another algorithm for computing the elementary divisors \(d_1, d_2, d_3, \ldots\), of a finite abelian group (where \(d_1 | d_2 | d_3 | \ldots\) are composed of prime powers dividing the invariants of the group in a way described below). Just factor the invariants \(a_i\) that define the abelian group. Then the biggest \(d_i\) is the product of the maximum prime powers dividing some \(a_j\). In other words, the largest \(d_i\) is the product of \(p^v\), where \(v = \max(\mathrm{ord}_p(a_j) \text{ for all } j\)). Now divide out all those \(p^v\)’s into the list of invariants \(a_i\), and get a new list of “smaller invariants”. Repeat the above procedure on these “smaller invariants” to compute \(d_{i-1}\), and so on. (Thanks to Robert Miller for communicating this algorithm.)

OUTPUT:

A tuple of integers.

EXAMPLES:

sage: G = AbelianGroup(2, [2,3])
sage: G.elementary_divisors()
(6,)
sage: G = AbelianGroup(1, [6])
sage: G.elementary_divisors()
(6,)
sage: G = AbelianGroup(2, [2,6])
sage: G
Multiplicative Abelian group isomorphic to C2 x C6
sage: G.gens_orders()
(2, 6)
sage: G.elementary_divisors()
(2, 6)
sage: J = AbelianGroup([1,3,5,12])
sage: J.elementary_divisors()
(3, 60)
sage: G = AbelianGroup(2, [0,6])
sage: G.elementary_divisors()
(6, 0)
sage: AbelianGroup([3,4,5]).elementary_divisors()
(60,)

exponent()#

Return the exponent of this abelian group.

EXAMPLES:

sage: G = AbelianGroup([2,3,7]); G
Multiplicative Abelian group isomorphic to C2 x C3 x C7
sage: G.exponent()
42
sage: G = AbelianGroup([2,4,6]); G
Multiplicative Abelian group isomorphic to C2 x C4 x C6
sage: G.exponent()
12

gen(i=0)#

The \(i\)-th generator of the abelian group.

EXAMPLES:

sage: F = AbelianGroup(5,[],names='a')
sage: F.0
a0
sage: F.2
a2
sage: F.gens_orders()
(0, 0, 0, 0, 0)

sage: G = AbelianGroup([2,1,3])
sage: G.gens()
(f0, 1, f2)

gens()#

Return the generators of the group.

OUTPUT:

A tuple of group elements. The generators according to the chosen gens_orders().

EXAMPLES:

sage: F = AbelianGroup(5, [3,2], names='abcde')
sage: F.gens()
(a, b, c, d, e)
sage: [g.order() for g in F.gens()]
[+Infinity, +Infinity, +Infinity, 3, 2]

gens_orders()#

Return the orders of the cyclic factors that this group has been defined with.

Use elementary_divisors() if you are looking for an invariant of the group.

OUTPUT:

A tuple of integers.

EXAMPLES:

sage: Z2xZ3 = AbelianGroup([2,3])
sage: Z2xZ3.gens_orders()
(2, 3)
sage: Z2xZ3.elementary_divisors()
(6,)

sage: Z6 = AbelianGroup([6])
sage: Z6.gens_orders()
(6,)
sage: Z6.elementary_divisors()
(6,)

sage: Z2xZ3.is_isomorphic(Z6)
True
sage: Z2xZ3 is Z6
False

identity()#

Return the identity element of this group.

EXAMPLES:

sage: G = AbelianGroup([2,2])
sage: e = G.identity()
sage: e
1
sage: g = G.gen(0)
sage: g*e
f0
sage: e*g
f0

invariants()#

Return the orders of the cyclic factors that this group has been defined with.

For historical reasons this has been called invariants in Sage, even though they are not necessarily the invariant factors of the group. Use gens_orders() instead:

sage: J = AbelianGroup([2,0,3,2,4]);  J
Multiplicative Abelian group isomorphic to C2 x Z x C3 x C2 x C4
sage: J.invariants()    # deprecated
(2, 0, 3, 2, 4)
sage: J.gens_orders()   # use this instead
(2, 0, 3, 2, 4)
sage: for i in range(J.ngens()):
....:     print((i, J.gen(i), J.gen(i).order()))     # or this
(0, f0, 2)
(1, f1, +Infinity)
(2, f2, 3)
(3, f3, 2)
(4, f4, 4)

Use elementary_divisors() if you are looking for an invariant of the group.

OUTPUT:

A tuple of integers. Zero means infinite cyclic factor.

EXAMPLES:

sage: J = AbelianGroup([2,3])
sage: J.invariants()
(2, 3)
sage: J.elementary_divisors()
(6,)

is_commutative()#

Return True since this group is commutative.

EXAMPLES:

sage: G = AbelianGroup([2,3,9, 0])
sage: G.is_commutative()
True
sage: G.is_abelian()
True

is_cyclic()#

Return True if the group is a cyclic group.

EXAMPLES:

sage: J = AbelianGroup([2,3])
sage: J.gens_orders()
(2, 3)
sage: J.elementary_divisors()
(6,)
sage: J.is_cyclic()
True
sage: G = AbelianGroup([6])
sage: G.gens_orders()
(6,)
sage: G.is_cyclic()
True
sage: H = AbelianGroup([2,2])
sage: H.gens_orders()
(2, 2)
sage: H.is_cyclic()
False
sage: H = AbelianGroup([2,4])
sage: H.elementary_divisors()
(2, 4)
sage: H.is_cyclic()
False
sage: H.permutation_group().is_cyclic()                                     # needs sage.groups
False
sage: T = AbelianGroup([])
sage: T.is_cyclic()
True
sage: T = AbelianGroup(1, [0]); T
Multiplicative Abelian group isomorphic to Z
sage: T.is_cyclic()
True
sage: B = AbelianGroup([3,4,5])
sage: B.is_cyclic()
True

is_isomorphic(left, right)#

Check whether left and right are isomorphic

INPUT:

right – anything.

OUTPUT:

Boolean. Whether left and right are isomorphic as abelian groups.

EXAMPLES:

sage: G1 = AbelianGroup([2,3,4,5])
sage: G2 = AbelianGroup([2,3,4,5,1])
sage: G1.is_isomorphic(G2)
True

is_subgroup(left, right)#

Test whether left is a subgroup of right.

EXAMPLES:

sage: G = AbelianGroup([2,3,4,5])
sage: G.is_subgroup(G)
True

sage: H = G.subgroup([G.1])                                                 # needs sage.libs.gap  # optional - gap_package_polycyclic
sage: H.is_subgroup(G)                                                      # needs sage.libs.gap  # optional - gap_package_polycyclic
True

sage: G.<a, b> = AbelianGroup(2)
sage: H.<c> = AbelianGroup(1)
sage: H < G
False

is_trivial()#

Return whether the group is trivial

A group is trivial if it has precisely one element.

EXAMPLES:

sage: AbelianGroup([2, 3]).is_trivial()
False
sage: AbelianGroup([1, 1]).is_trivial()
True

list()#

Return tuple of all elements of this group.

EXAMPLES:

sage: G = AbelianGroup([2,3], names = "ab")
sage: G.list()
(1, b, b^2, a, a*b, a*b^2)

sage: G = AbelianGroup([]); G
Trivial Abelian group
sage: G.list()
(1,)

ngens()#

The number of free generators of the abelian group.

EXAMPLES:

sage: F = AbelianGroup(10000)
sage: F.ngens()
10000

number_of_subgroups(order=None)#

Return the number of subgroups of this group, possibly only of a specific order.

INPUT:

order – (default: None) find the number of subgroups of this order; if None, this defaults to counting all subgroups

ALGORITHM:

An infinite group has infinitely many subgroups. All finite subgroups of any group are contained in the torsion subgroup, which for finitely generated abelian group is itself finite. Hence, we can assume the group is finite. A finite abelian group is isomorphic to a direct product of its Sylow subgroups, and so we can reduce the problem further to counting subgroups of finite abelian \(p\)-groups.

Assume a Sylow subgroup is a \(p\)-group of type \(\lambda\), and using q_subgroups_of_abelian_group() sum the number of subgroups of type \(\mu\) in an abelian \(p\)-group of type \(\lambda\) for all \(\mu\) contained in \(\lambda\).

EXAMPLES:

sage: AbelianGroup([2,0,0,3,0]).number_of_subgroups()
+Infinity

sage: # needs sage.combinat
sage: AbelianGroup([2,3]).number_of_subgroups()
4
sage: AbelianGroup([2,4,8]).number_of_subgroups()
81
sage: AbelianGroup([2,4,8]).number_of_subgroups(order=4)
19
sage: AbelianGroup([10,15,25,12]).number_of_subgroups()
5760
sage: AbelianGroup([10,15,25,12]).number_of_subgroups(order=45000)
1
sage: AbelianGroup([10,15,25,12]).number_of_subgroups(order=14)
0

order()#

Return the order of this group.

EXAMPLES:

sage: G = AbelianGroup(2, [2,3])
sage: G.order()
6
sage: G = AbelianGroup(3, [2,3,0])
sage: G.order()
+Infinity

permutation_group()#

Return the permutation group isomorphic to this abelian group.

If the invariants are \(q_1, \ldots, q_n\) then the generators of the permutation will be of order \(q_1, \ldots, q_n\), respectively.

EXAMPLES:

sage: G = AbelianGroup(2,[2,3]); G
Multiplicative Abelian group isomorphic to C2 x C3
sage: G.permutation_group()                                                 # needs sage.groups
Permutation Group with generators [(3,4,5), (1,2)]

random_element()#

Return a random element of this group.

EXAMPLES:

sage: G = AbelianGroup([2,3,9])
sage: G.random_element().parent() is G
True

subgroup(gensH, names='f')#

Create a subgroup of this group.

The “big” group must be defined using “named” generators.

INPUT:

gensH – list of elements which are products of the generators of the ambient abelian group \(G\) = self

EXAMPLES:

sage: # needs sage.libs.gap  # optional - gap_package_polycyclic
sage: G.<a,b,c> = AbelianGroup(3, [2,3,4]); G
Multiplicative Abelian group isomorphic to C2 x C3 x C4
sage: H = G.subgroup([a*b,a]); H
Multiplicative Abelian subgroup isomorphic to C2 x C3 generated by {a*b, a}
sage: H < G
True
sage: F = G.subgroup([a,b^2])
sage: F
Multiplicative Abelian subgroup isomorphic to C2 x C3 generated by {a, b^2}
sage: F.gens()
(a, b^2)
sage: F = AbelianGroup(5, [30,64,729], names=list("abcde"))
sage: a,b,c,d,e = F.gens()
sage: F.subgroup([a,b])
Multiplicative Abelian subgroup isomorphic to Z x Z generated by {a, b}
sage: F.subgroup([c,e])
Multiplicative Abelian subgroup isomorphic to C2 x C3 x C5 x C729
generated by {c, e}

subgroup_reduced(elts, verbose=False)#

Given a list of lists of integers (corresponding to elements of self), find a set of independent generators for the subgroup generated by these elements, and return the subgroup with these as generators, forgetting the original generators.

This is used by the subgroups routine.

An error will be raised if the elements given are not linearly independent over QQ.

EXAMPLES:

sage: G = AbelianGroup([4,4])
sage: G.subgroup( [ G([1,0]), G([1,2]) ])                                   # needs sage.libs.gap  # optional - gap_package_polycyclic
Multiplicative Abelian subgroup isomorphic to C2 x C4
generated by {f0, f0*f1^2}
sage: AbelianGroup([4,4]).subgroup_reduced( [ [1,0], [1,2] ])               # needs sage.libs.gap  # optional - gap_package_polycyclic
Multiplicative Abelian subgroup isomorphic to C2 x C4
generated by {f0^2*f1^2, f0^3}

subgroups(check=False)#

Compute all the subgroups of this abelian group (which must be finite).

INPUT:

check: if True, performs the same computation in GAP and checks that the number of subgroups generated is the same. (I don’t know how to convert GAP’s output back into Sage, so we don’t actually compare the subgroups).

ALGORITHM:

If the group is cyclic, the problem is easy. Otherwise, write it as a direct product A x B, where B is cyclic. Compute the subgroups of A (by recursion).

Now, for every subgroup C of A x B, let G be its projection onto A and H its intersection with B. Then there is a well-defined homomorphism f: G -> B/H that sends a in G to the class mod H of b, where (a,b) is any element of C lifting a; and every subgroup C arises from a unique triple (G, H, f).

Todo

This is many orders of magnitude slower than Magma. Consider using the much faster method number_of_subgroups() in case you only need the number of subgroups, possibly of a specific order.

EXAMPLES:

sage: AbelianGroup([2,3]).subgroups()                                       # needs sage.libs.gap  # optional - gap_package_polycyclic
[Multiplicative Abelian subgroup isomorphic to C2 x C3 generated by {f0*f1^2},
 Multiplicative Abelian subgroup isomorphic to C2 generated by {f0},
 Multiplicative Abelian subgroup isomorphic to C3 generated by {f1},
 Trivial Abelian subgroup]
sage: len(AbelianGroup([2,4,8]).subgroups())                                # needs sage.libs.gap  # optional - gap_package_polycyclic
81

torsion_subgroup(n=None)#

Return the \(n\)-torsion subgroup of this abelian group when \(n\) is given, and the torsion subgroup otherwise.

The [\(n\)-]torsion subgroup consists of all elements whose order is finite [and divides \(n\)].

EXAMPLES:

sage: # needs sage.libs.gap  # optional - gap_package_polycyclic
sage: G = AbelianGroup([2, 3])
sage: G.torsion_subgroup()
Multiplicative Abelian subgroup isomorphic to C2 x C3 generated
by {f0, f1}
sage: G = AbelianGroup([2, 0, 0, 3, 0])
sage: G.torsion_subgroup()
Multiplicative Abelian subgroup isomorphic to C2 x C3 generated
by {f0, f3}
sage: G = AbelianGroup([])
sage: G.torsion_subgroup()
Trivial Abelian subgroup
sage: G = AbelianGroup([0, 0])
sage: G.torsion_subgroup()
Trivial Abelian subgroup

sage: G = AbelianGroup([2, 2*3, 2*3*5, 0, 2*3*5*7, 2*3*5*7*11])
sage: G.torsion_subgroup(5)                                                 # needs sage.libs.gap  # optional - gap_package_polycyclic
Multiplicative Abelian subgroup isomorphic to C5 x C5 x C5 generated by {f2^6, f4^42, f5^462}

class sage.groups.abelian_gps.abelian_group.AbelianGroup_subgroup(ambient, gens, names='f', category=None)#

Bases: AbelianGroup_class

Subgroup subclass of AbelianGroup_class, so instance methods are inherited.

Todo

There should be a way to coerce an element of a subgroup into the ambient group.

ambient_group()#

Return the ambient group related to self.

OUTPUT:

A multiplicative Abelian group.

EXAMPLES:

sage: # needs sage.libs.gap  # optional - gap_package_polycyclic
sage: G.<a,b,c> = AbelianGroup([2,3,4])
sage: H = G.subgroup([a, b^2])
sage: H.ambient_group() is G
True

equals(left, right)#

Check whether left and right are the same (sub)group.

INPUT:

right – anything.

OUTPUT:

Boolean. If right is a subgroup, test whether left and right are the same subset of the ambient group. If right is not a subgroup, test whether they are isomorphic groups, see is_isomorphic().

EXAMPLES:

sage: # needs sage.libs.gap  # optional - gap_package_polycyclic
sage: G = AbelianGroup(3, [2,3,4], names="abc"); G
Multiplicative Abelian group isomorphic to C2 x C3 x C4
sage: a,b,c = G.gens()
sage: F = G.subgroup([a,b^2]); F
Multiplicative Abelian subgroup isomorphic to C2 x C3 generated by {a, b^2}
sage: F<G
True
sage: A = AbelianGroup(1, [6])
sage: A.subgroup(list(A.gens())) == A
True
sage: G.<a,b> = AbelianGroup(2)
sage: A = G.subgroup([a])
sage: B = G.subgroup([b])
sage: A.equals(B)
False
sage: A == B        # sames as A.equals(B)
False
sage: A.is_isomorphic(B)
True

gen(n)#

Return the nth generator of this subgroup.

EXAMPLES:

sage: # needs sage.libs.gap  # optional - gap_package_polycyclic
sage: G.<a,b> = AbelianGroup(2)
sage: A = G.subgroup([a])
sage: A.gen(0)
a

gens()#

Return the generators for this subgroup.

OUTPUT:

A tuple of group elements generating the subgroup.

EXAMPLES:

sage: # needs sage.libs.gap  # optional - gap_package_polycyclic
sage: G.<a,b> = AbelianGroup(2)
sage: A = G.subgroup([a])
sage: G.gens()
(a, b)
sage: A.gens()
(a,)

sage.groups.abelian_gps.abelian_group.is_AbelianGroup(x)#

Return True if x is an Abelian group.

EXAMPLES:

sage: from sage.groups.abelian_gps.abelian_group import is_AbelianGroup
sage: F = AbelianGroup(5,[5,5,7,8,9], names=list("abcde")); F
Multiplicative Abelian group isomorphic to C5 x C5 x C7 x C8 x C9
sage: is_AbelianGroup(F)
True
sage: is_AbelianGroup(AbelianGroup(7, [3]*7))
True

sage.groups.abelian_gps.abelian_group.word_problem(words, g, verbose=False)#

G and H are abelian, g in G, H is a subgroup of G generated by a list (words) of elements of G. If g is in H, return the expression for g as a word in the elements of (words).

The ‘word problem’ for a finite abelian group G boils down to the following matrix-vector analog of the Chinese remainder theorem.

Problem: Fix integers \(1<n_1\leq n_2\leq ...\leq n_k\) (indeed, these \(n_i\) will all be prime powers), fix a generating set \(g_i=(a_{i1},...,a_{ik})\) (with \(a_{ij}\in \mathrm{Z}/n_j\mathrm{Z}\)), for \(1\leq i\leq \ell\), for the group \(G\), and let \(d = (d_1,...,d_k)\) be an element of the direct product \(\mathrm{Z}/n_1\mathrm{Z} \times ...\times \mathrm{Z}/n_k\mathrm{Z}\). Find, if they exist, integers \(c_1,...,c_\ell\) such that \(c_1g_1+...+c_\ell g_\ell = d\). In other words, solve the equation \(cA=d\) for \(c\in \mathrm{Z}^\ell\), where \(A\) is the matrix whose rows are the \(g_i\)’s. Of course, it suffices to restrict the \(c_i\)’s to the range \(0\leq c_i\leq N-1\), where \(N\) denotes the least common multiple of the integers \(n_1,...,n_k\).

This function does not solve this directly, as perhaps it should. Rather (for both speed and as a model for a similar function valid for more general groups), it pushes it over to GAP, which has optimized (non-deterministic) algorithms for the word problem. Essentially, this function is a wrapper for the GAP function ‘Factorization’.

EXAMPLES:

sage: # needs sage.libs.gap
sage: G.<a,b,c> = AbelianGroup(3, [2,3,4]); G
Multiplicative Abelian group isomorphic to C2 x C3 x C4
sage: w = word_problem([a*b,a*c], b*c); w       # random
[[a*b, 1], [a*c, 1]]
sage: prod([x^i for x,i in w]) == b*c
True
sage: w = word_problem([a*c,c], a); w           # random
[[a*c, 1], [c, -1]]
sage: prod([x^i for x,i in w]) == a
True
sage: word_problem([a*c,c], a, verbose=True)    # random
a = (a*c)^1*(c)^-1
[[a*c, 1], [c, -1]]

sage: # needs sage.libs.gap
sage: A.<a,b,c,d,e> = AbelianGroup(5, [4, 5, 5, 7, 8])
sage: b1 = a^3*b*c*d^2*e^5
sage: b2 = a^2*b*c^2*d^3*e^3
sage: b3 = a^7*b^3*c^5*d^4*e^4
sage: b4 = a^3*b^2*c^2*d^3*e^5
sage: b5 = a^2*b^4*c^2*d^4*e^5
sage: w = word_problem([b1,b2,b3,b4,b5], e); w  # random
[[a^3*b*c*d^2*e^5, 1], [a^2*b*c^2*d^3*e^3, 1],
 [a^3*b^3*d^4*e^4, 3], [a^2*b^4*c^2*d^4*e^5, 1]]
sage: prod([x^i for x,i in w]) == e
True
sage: word_problem([a,b,c,d,e], e)
[[e, 1]]
sage: word_problem([a,b,c,d,e], b)
[[b, 1]]

Warning

Might have unpleasant effect when the word problem cannot be solved.
Uses permutation groups, so may be slow when group is large. The instance method word_problem of the class AbelianGroupElement is implemented differently (wrapping GAP’s ‘EpimorphismFromFreeGroup’ and ‘PreImagesRepresentative’) and may be faster.