7.100 SemidirectProduct

SemidirectProduct( G, a, H )

SemidirectProduct returns the semidirect product of G with H. a must be a homomorphism that from G onto a group A that operates on H via the caret (^) operator. A may either be a subgroup of the parent group of H that normalizes H, or a subgroup of the Group Homomorphisms).

The semidirect product of G and H is a the group of pairs (g,h) with g in G and h in H, where the product of (g_1,h_1) (g_2,h_2) is defined as (g_1 g_2, h_1^{g_2^a} h_2). Note that the elements (1_G,h) form a normal subgroup in the semidirect product.

Embedding( U, S, 1 )

Let U be a subgroup of G. Embedding returns the homomorphism of U into the semidirect product S where u is mapped to (u,1).

Embedding( U, S, 2 )

Let U be a subgroup of H. Embedding returns the homomorphism of U into the semidirect product S where u is mapped to (1,u).

Projection( S, G, 1 )

Projection returns the homomorphism of S onto G, where (g,h) is mapped to g.

Projection( S, H, 2 )

Projection returns the homomorphism of S onto H, where (g,h) is mapped to h.

It is not specified how the elements of the semidirect product are represented. Thus Embedding and Projection are the only general possibility to relate G and H with the semidirect product.

    gap> s4 := Group( (1,2), (1,2,3,4) );;  s4.name := "s4";;
    gap> s3 := Subgroup( s4, [ (1,2), (1,2,3) ] );; s3.name := "s3";;
    gap> a4 := Subgroup( s4, [ (1,2,3), (2,3,4) ] );;  a4.name := "a4";;
    gap> a := IdentityMapping( s3 );;
    gap> s := SemidirectProduct( s3, a, a4 );
    Group( SemidirectProductElement( (1,2),
    (1,2), () ), SemidirectProductElement( (1,2,3),
    (1,2,3), () ), SemidirectProductElement( (), (),
    (1,2,3) ), SemidirectProductElement( (), (), (2,3,4) ) )
    gap> Size( s );
    72 

Note that the three arguments of SemidirectProductElement are the element g, its image under a, and the element h.

SemidirectProduct calls the function G.operations.SemidirectProduct with the arguments G, a, and H, and returns the result.

The default function called this way is GroupOps.SemidirectProduct. This function constructs the semidirect product as a group of semidirect product elements (see SemidirectProduct for Groups). Look in the index under SemidirectProduct to see for which groups this function is overlaid.

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