Coverage for local/lib/python2.7/site-packages/sage/groups/abelian_gps/abelian_group_element.py : 72%

""" 

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] 

""" 

########################################################################### 

# Copyright (C) 2006 William Stein <wstein@gmail.com> 

# Copyright (C) 2006 David Joyner <wdjoyner@gmail.com> 

# Copyright (C) 2012 Volker Braun <vbraun.name@gmail.com> 

# 

# Distributed under the terms of the GNU General Public License (GPL) 

# http://www.gnu.org/licenses/ 

########################################################################### 

from sage.rings.integer import Integer 

from sage.rings.infinity import infinity 

from sage.arith.all import LCM, GCD 

from sage.groups.abelian_gps.element_base import AbelianGroupElementBase 

def 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 

""" 

return isinstance(x, AbelianGroupElement) 

class AbelianGroupElement(AbelianGroupElementBase): 

""" 

Elements of an 

:class:`~sage.groups.abelian_gps.abelian_group.AbelianGroup` 

INPUT: 

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

- ``parent`` -- the parent 

:class:`~sage.groups.abelian_gps.abelian_group.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 

""" 

def as_permutation(self): 

r""" 

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 

""" 

from sage.groups.perm_gps.permgroup import PermutationGroup 

from sage.interfaces.all import gap 

G = self.parent() 

invs = list(G.gens_orders()) 

s1 = 'A:=AbelianGroup(%s)'%invs 

gap.eval(s1) 

s2 = 'phi:=IsomorphismPermGroup(A)' 

gap.eval(s2) 

s3 = "gens := GeneratorsOfGroup(A)" 

gap.eval(s3) 

L = self.list() 

gap.eval("L1:="+str(L)) 

s4 = "L2:=List([1..%s], i->gens[i]^L1[i]);"%len(L) 

gap.eval(s4) 

pg = gap.eval("Image(phi,Product(L2))") 

Gp = G.permutation_group() 

gp = Gp(pg) 

return gp 

def word_problem(self, 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 

""" 

from sage.groups.abelian_gps.abelian_group import AbelianGroup, word_problem 

return word_problem(words,self)