8.19 OperationHomomorphism

OperationHomomorphism( G, P )

Group Homomorphisms) from the group G to the permutation group P, which must be the result of a prior call to Operation (see Operation) with G or a group of which G is a subgroup (see IsSubgroup) as first argument.

    gap> g := Group( (1,2,3)(6,7), (3,4,5)(7,8) );;
    gap> h := Operation( g, [1..5] );
    Group( (1,2,3), (3,4,5) )
    gap> p := OperationHomomorphism( g, h );
    OperationHomomorphism( Group( (1,2,3)(6,7), (3,4,5)(7,8) ), Group(
    (1,2,3), (3,4,5) ) )
    gap> (1,4,2,5,3)(6,7,8) ^ p;
    (1,4,2,5,3)
    gap> h := Operation( g, Orbit( g, [1,6], OnPairs ), OnPairs );
    Group( ( 1, 2, 3, 5, 8,12)( 4, 7, 9)( 6,10)(11,14), ( 2, 4)( 3, 6,11)
    ( 5, 9)( 7,10,13,12,15,14) )
    gap> p := OperationHomomorphism( g, h );;
    gap> s := SylowSubgroup( g, 2 );
    Subgroup( Group( (1,2,3)(6,7), (3,4,5)(7,8) ),
    [ (7,8), (7,8), (2,5)(3,4), (2,3)(4,5) ] )
    gap> Images( p, s );
    Subgroup( Group( ( 1, 2, 3, 5, 8,12)( 4, 7, 9)( 6,10)(11,14), ( 2, 4)
    ( 3, 6,11)( 5, 9)( 7,10,13,12,15,14) ),
    [ ( 2, 4)( 5, 9)( 7,12)(10,15)(13,14),
      ( 2, 4)( 5, 9)( 7,12)(10,15)(13,14),
      ( 2,14)( 3, 6)( 4,13)( 7,15)( 8,11)(10,12),
      ( 2,12)( 3, 8)( 4, 7)( 6,11)(10,14)(13,15) ] )
    gap> OperationHomomorphism( g, Group( (1,2,3), (3,4,5) ) );
    Error, Record: element 'operation' must have an assigned value 

OperationHomomorphism calls
P.operations.OperationHomomorphism( G, P )
and returns the value.

The default function called this way is GroupOps.OperationHomomorphism, which uses the fields P.operationGroup, P.operationDomain, and P.operationOperation (the arguments to the Operation call that created P) to construct a generic homomorphism h. This homomorphism uses
Permutation(g,h.range.operationDomain,h.range.operationOperation)
to compute the image of an element g of G under h. It uses Representative to compute the preimages of an element p of P under h. And it computes the kernel by intersecting the cores (see Core) of the stabilizers (see Stabilizer) of representatives of the orbits of G. Look under OperationHomomorphism in the index to see for which groups and operations this function is overlaid.

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