GAP Manual: 7.98. DirectProduct

DirectProduct( G_1, ..., G_n )

DirectProduct returns a group record of the direct product D of the groups G_1, ...., G_n which need not to have a common parent group, it is even possible to construct the direct product of an ag group with a permutation group.

Note that the elements of the direct product may be just represented as records. But more complicate constructions, as for instance installing a new collector, may be used. The choice of method strongly depends on the type of group arguments.

Embedding( U, D, i )

Let U be a subgroup of G_i. Embedding returns a homomorphism of U into D which describes the embedding of U in D.

Projection( D, U, i )

Let U be a supergroup of G_i. Projection returns a homomorphism of D into U which describes the projection of D onto G_i.

    gap> s4 := Group( (1,2,3,4), (1,2) );
    Group( (1,2,3,4), (1,2) )
    gap> S4 := AgGroup( s4 );
    Group( g1, g2, g3, g4 )
    gap> D := DirectProduct( s4, S4 );
    Group( DirectProductElement( (1,2,3,4), IdAgWord ),
      DirectProductElement( (1,2), IdAgWord ),
      DirectProductElement( (), g1 ),
      DirectProductElement( (), g2 ),
      DirectProductElement( (),
    g3 ), DirectProductElement( (), g4 ) )
    gap> pr := Projection( D, s4, 1 );;
    gap> Image( pr );
    Group( (1,2,3,4), (1,2) )