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Dear gap-forum,

In answer to Kurt Ewald's question (see beelow):

Given two groups G and H in GAP, there is a difference between the

statement that H is a subgroup of G, and the statement that G has

a subgroup isomorphic to H. To find z4 in your p, you can use

Embedding(p,1);

this gives a monomorphism z4->p. To find z3, you can use

Embedding(p,2);

To find the projection of p onto z4, use

Projection(p);

Best wishes,

Jan > Dear forum > theory says that SmallGroup(12,1) is the semidirect product of z3 by z4 > indeed > gap> z3;z4; > Group([ (1,2,3) ]) > Group([ (1,2,3,4) ]) > gap> aut:=AutomorphismGroup(z3); > <group of size 2 with 1 generators> > hom:=GroupHomomorphismByImages(z4,aut,[(1,2,3,4)],[aut.1]); > gap> p:=SemidirectProduct(z4,hom,z3); > Group([ (2,3)(4,5,6,7), (1,2,3) ]) > gap> x:=SmallGroup(12,1); > <pc group of size 12 with 3 generators> > gap> IsomorphismGroups(p,x); > [ (2,3)(4,5,6,7), (1,3,2) ] -> [ f1, f3 ] > But > gap> IsSubgroup(p,z4); > false > gap> IsSubgroup(x,z4); > false > The complement (z4) of z3 must be a subgroup of the semiproduct. > Where lies the error? > Best wishes > K. Ewald > >

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