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14.3 Abelian group elements

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= 14.3 Abelian group elements =
<<Anchor(SECTION0016300000000000000000)>>
 '''Module:''' {{{sage.groups.abelian_gps.abelian_group_element}}}

<<Anchor(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 order<<BR>>

 * " (2006-08); bug fix+new method in pow for negatives+fixed corresponding
 examples.<<BR>>

Recall an example from abelian groups.
{{{#!python
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]
}}}

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.