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]

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

Bases: AbelianGroupElementBase

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

>>> from sage.all import *
>>> F = AbelianGroup(Integer(5), [Integer(3),Integer(4),Integer(5),Integer(8),Integer(7)], 'abcde')
>>> a, b, c, d, e = F.gens()
>>> a**Integer(2) * b**Integer(3) * a**Integer(2) * b**-Integer(4)
a*b^3
>>> b**-Integer(11)
b
>>> a**-Integer(11)
a
>>> a*b in F
True
as_permutation()[source]#

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                                        # needs sage.groups
Permutation Group with generators [(1,2), (3,4,5), (6,7,8,9)]
sage: a.as_permutation()                                                    # needs sage.libs.gap
(1,2)
sage: ap = a.as_permutation(); ap                                           # needs sage.libs.gap
(1,2)
sage: ap in Gp                                                              # needs sage.groups sage.libs.gap
True

>>> from sage.all import *
>>> G = AbelianGroup(Integer(3), [Integer(2),Integer(3),Integer(4)], names="abc"); G
Multiplicative Abelian group isomorphic to C2 x C3 x C4
>>> a,b,c = G.gens()
>>> Gp = G.permutation_group(); Gp                                        # needs sage.groups
Permutation Group with generators [(1,2), (3,4,5), (6,7,8,9)]
>>> a.as_permutation()                                                    # needs sage.libs.gap
(1,2)
>>> ap = a.as_permutation(); ap                                           # needs sage.libs.gap
(1,2)
>>> ap in Gp                                                              # needs sage.groups sage.libs.gap
True
word_problem(words)[source]#

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.

Todo

This needs a rewrite - see stuff in the matrix_grp directory.

EXAMPLES:

sage: # needs sage.libs.gap
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

>>> from sage.all import *
>>> # needs sage.libs.gap
>>> G = AbelianGroup(Integer(2), [Integer(2),Integer(3)], names="xy")
>>> x,y = G.gens()
>>> x.word_problem([x,y])
[[x, 1]]
>>> y.word_problem([x,y])
[[y, 1]]
>>> v = (y*x).word_problem([x,y]); v  # random
[[x, 1], [y, 1]]
>>> prod([x**i for x,i in v]) == y*x
True
sage.groups.abelian_gps.abelian_group_element.is_AbelianGroupElement(x)[source]#

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)
doctest:warning...
DeprecationWarning: The function is_AbelianGroupElement is deprecated;
use 'isinstance(..., AbelianGroupElement)' instead.
See https://github.com/sagemath/sage/issues/38184 for details.
False
sage: F = AbelianGroup(5, [3,4,5,8,7], 'abcde')
sage: is_AbelianGroupElement(F.0)
True

>>> from sage.all import *
>>> from sage.groups.abelian_gps.abelian_group_element import is_AbelianGroupElement
>>> is_AbelianGroupElement(Integer(3))
doctest:warning...
DeprecationWarning: The function is_AbelianGroupElement is deprecated;
use 'isinstance(..., AbelianGroupElement)' instead.
See https://github.com/sagemath/sage/issues/38184 for details.
False
>>> F = AbelianGroup(Integer(5), [Integer(3),Integer(4),Integer(5),Integer(8),Integer(7)], 'abcde')
>>> is_AbelianGroupElement(F.gen(0))
True