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)#
Bases:
AbelianGroupElementBase
Elements of an
AbelianGroup
INPUT:
x
– list/tuple/iterable of integers (the element vector)parent
– the parentAbelianGroup
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 # needs sage.groups Permutation Group with generators [(6,7,8,9), (3,4,5), (1,2)] 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
- 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: # 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
- 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