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