36.17 Factor Groups

returns a homomorphism from G to another group whose kernel is N. GAP will try to select the image group as to make computations in it as efficient as possible. As the factor group G /N can be identified with the image of G this permits efficient computations in the factor group. The homomorphism returned is not necessarily surjective, so ImagesSource should be used instead of Range to get a group isomorphic to the factor group. The NC variant does not check whether N is normal in G.

returns the image of the NaturalHomomorphismByNormalSubgroup(G,N). The NC version does not test whether N is normal in G.

gap> g:=Group((1,2,3,4),(1,2));;n:=Subgroup(g,[(1,2)(3,4),(1,3)(2,4)]);;
gap> hom:=NaturalHomomorphismByNormalSubgroup(g,n);
[ (3,4), (2,4,3), (1,2)(3,4), (1,3)(2,4) ] -> 
[ f1, f2, <identity> of ..., <identity> of ... ]
gap> Size(ImagesSource(hom));
6
gap> FactorGroup(g,n);
<pc group of size 6 with 2 generators>

computes the commutator factor group G /G ^¢ of the group G.

gap> CommutatorFactorGroup(g);
<pc group of size 2 with 1 generators>

tests whether G/N is abelian (without explicitly constructing the factor group).

tests whether G/N is elementary abelian (without explicitly constructing the factor group).

gap> HasAbelianFactorGroup(g,n);
false
gap> HasAbelianFactorGroup(DerivedSubgroup(g),n);  
true

Computes the full preimage of the centralizer C_G/N(elm·N) in G (without necessarily constructing the factor group).

gap> CentralizerModulo(g,n,(1,2));
Group( [ (3,4), (1,4)(2,3), (1,3)(2,4) ] )

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