# 25.21 WreathProduct for Ag Groups

AgGroupOps.WreathProduct( G, H, alpha )

If H and G are not both ag group GroupOps.WreathProduct (see WreathProduct) is used.

Let H and G be two ag group with possible different parent group and let alpha be a homomorphism H into a permutation group of degree d.

Let (g_1, ..., g_r) be an IGS of G, (h_1, ..., h_n) an IGS of H. The wreath product has a PAG system (b_1, ..., b_n, a_{11}, ..., a_{1r}, a_{d1}, ..., a_{dr}) such that b_1, ..., b_n generate a subgroup isomorph to H and a_{i1}, ..., a_{ir} generate a subgroup isomorph to G for each i in {1, ..., r}. The names of b_1, ..., b_n are h1, ..., hn, the names of a_{i1}, ..., a_{ir} are ni_1, ..., ni_r.

AgGroupOps.WreathProduct uses the natural power-commutator presentations of H and G for induced generating system of H and G (see Thi87).

gap> s3 := Subgroup( s4, [ a, b ] );
Subgroup( s4, [ a, b ] )
gap> c2 := Subgroup( s4, [ a ] );
Subgroup( s4, [ a ] )
gap> r := RightCosets( s3, c2 );;
gap> S3 := Operation( s3, r, OnRight );
Group( (2,3), (1,2,3) )
gap> f := GroupHomomorphismByImages(s3,S3,[a,b],[(2,3),(1,2,3)]);
GroupHomomorphismByImages( Subgroup( s4, [ a, b ] ), Group( (2,3),
(1,2,3) ), [ a, b ], [ (2,3), (1,2,3) ] )
gap> WreathProduct( c2, s3, f );
Group( h1, h2, n1_1, n2_1, n3_1 )

GAP 3.4.4
April 1997