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14.3 Abelian group elements

 [http://modular.math.washington.edu/sage/doc/html/ref/contents.html SAGE Reference Manual]

# 14.3 Abelian group elements

• Module: sage.groups.abelian_gps.abelian_group_element

• Author Log: BR

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

by David Kohel.BR

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

• " (2006-08); bug fix+new method in pow for negatives+fixed corresponding

examples.BR

Recall an example from abelian groups.

```   1 sage: F = AbelianGroup(5,[4,5,5,7,8],names = list("abcde"))
2 sage: (a,b,c,d,e) = F.gens()
3 sage: x = a*b^2*e*d^20*e^12
4 sage: x
5 a*b^2*d^6*e^5
6 sage: x = a^10*b^12*c^13*d^20*e^12
7 sage: x
8 a^2*b^2*c^3*d^6*e^4
9 sage: y = a^13*b^19*c^23*d^27*e^72
10 sage: y
11 a*b^4*c^3*d^6
12 sage: x*y
13 a^3*b*c*d^5*e^4
14 sage: x.list()
15 [2, 2, 3, 6, 4]
```

It is important to note that lists are mutable and the returned list is not a copy. As a result, reassignment of an element of the list changes the object.

```sage: x.list()[0] = 3
sage: x.list()
[3, 2, 3, 6, 4]
sage: x
a^3*b^2*c^3*d^6*e^4```
• Module-level Functions

is_AbelianGroupElement (x )

Class: AbelianGroupElement

class AbelianGroupElement AbelianGroupElement (self, F, x )

Create the element x of the AbelianGroup F.

```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```
• Functions: as_permutation , list , order , random , word_problem

as_permutation (self )

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