# Abelian group elements¶

AUTHORS:

• David Joyner (2006-02); based on free_abelian_monoid_element.py, written by David Kohel.

• David Joyner (2006-05); bug fix in order

• David Joyner (2006-08); bug fix+new method in pow for negatives+fixed corresponding examples.

• David Joyner (2009-02): Fixed bug in order.

• Volker Braun (2012-11) port to new Parent base. Use tuples for immutables.

EXAMPLES:

Recall an example from abelian groups:

```sage: F = AbelianGroup(5,[4,5,5,7,8],names = list("abcde"))
sage: (a,b,c,d,e) = F.gens()
sage: x = a*b^2*e*d^20*e^12
sage: x
a*b^2*d^6*e^5
sage: x = a^10*b^12*c^13*d^20*e^12
sage: x
a^2*b^2*c^3*d^6*e^4
sage: y = a^13*b^19*c^23*d^27*e^72
sage: y
a*b^4*c^3*d^6
sage: x*y
a^3*b*c*d^5*e^4
sage: x.list()
[2, 2, 3, 6, 4]
```
class sage.groups.abelian_gps.abelian_group_element.AbelianGroupElement(parent, exponents)

Elements of an `AbelianGroup`

INPUT:

• `x` – list/tuple/iterable of integers (the element vector)

• `parent` – the parent `AbelianGroup`

EXAMPLES:

```sage: F = AbelianGroup(5, [3,4,5,8,7], 'abcde')
sage: a, b, c, d, e = F.gens()
sage: a^2 * b^3 * a^2 * b^-4
a*b^3
sage: b^-11
b
sage: a^-11
a
sage: a*b in F
True
```
as_permutation()

Return the element of the permutation group G (isomorphic to the abelian group A) associated to a in A.

EXAMPLES:

```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: Gp = G.permutation_group(); Gp
Permutation Group with generators [(6,7,8,9), (3,4,5), (1,2)]
sage: a.as_permutation()
(1,2)
sage: ap = a.as_permutation(); ap
(1,2)
sage: ap in Gp
True
```
word_problem(words)

TODO - this needs a rewrite - see stuff in the matrix_grp directory.

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

This function does not solve the word problem in Sage. Rather it pushes it over to GAP, which has optimized (non-deterministic) algorithms for the word problem.

Warning

Don’t use E (or other GAP-reserved letters) as a generator name.

EXAMPLES:

```sage: G = AbelianGroup(2,[2,3], names="xy")
sage: x,y = G.gens()
sage: x.word_problem([x,y])
[[x, 1]]
sage: y.word_problem([x,y])
[[y, 1]]
sage: v = (y*x).word_problem([x,y]); v #random
[[x, 1], [y, 1]]
sage: prod([x^i for x,i in v]) == y*x
True
```
sage.groups.abelian_gps.abelian_group_element.is_AbelianGroupElement(x)

Return true if x is an abelian group element, i.e., an element of type AbelianGroupElement.

EXAMPLES: Though the integer 3 is in the integers, and the integers have an abelian group structure, 3 is not an AbelianGroupElement:

```sage: from sage.groups.abelian_gps.abelian_group_element import is_AbelianGroupElement
sage: is_AbelianGroupElement(3)
False
sage: F = AbelianGroup(5, [3,4,5,8,7], 'abcde')
sage: is_AbelianGroupElement(F.0)
True
```