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Dear GAP Forum,

Kurt Ewald asked for the meaning of `IsSubgroup'.

As the Reference Manual says,

`IsSubgroup( <G>, <U> )' is `true' if <G> and <U>

are groups such that <U> is a subset of <G>.

The manual entry for `IsSubset' implies that the

relation checked by `IsSubgroup' is stronger than the

relation ``<U> is isomorphic to a subgroup of <G>''.

As for the given example of semidirect products,

the manual entry for `SemidirectProduct' says

that one has to use explicit embeddings

when one wants to access the normal subgroup

and a complement, respectively,

as subgroups of the semidirect product.

The example might be treated as follows.

gap> z3:= Group( (1,2,3) );; z4:= Group( (1,2,3,4) );; gap> aut:= AutomorphismGroup( z3 );; gap> 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> emb:= Embedding( p, 1 ); [ (1,2,3,4) ] -> [ (2,3)(4,5,6,7) ] gap> IsSubgroup( p, Image( emb, z4 ) ); true

Kind regards,

Thomas Breuer

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