70.8 Isomorphisms

Isomorphisms( P1, P2 )

If the polycyclically or freely presented p-groups P1 and the polycyclically presented p-group P2 are not isomorphic, Isomorphisms returns false. Otherwise a record is returned that encodes the isomorphisms from P1 to P2; its components are

epimorphism:

a list of images of P1.generators that defines an isomorphism from P1 to P2,

generators:

a list of image lists which encode automorphisms that together with the inner automorphisms generate the full automorphism group of P2

sizeOutG:

size of the group of outer automorphisms of P2,

sizeInnG:

size of the group of inner automorphisms of P2,

sizeOutG:

size of the full automorphism group of P2.

    gap> g:= SolvableGroup( "Q8" );;
    gap> nonab:= AllTwoGroups( Size, 8, IsAbelian, false );
    [ Group( a1, a2, a3 ), Group( a1, a2, a3 ) ]
    gap> nonab[2].name:= "im";;
    gap> Isomorphisms( g, nonab[2] );
    rec(
    sizeAutG := 24,
    sizeInnG := 4,
    sizeOutG := 6,
    epimorphism := [ a1, a2, a3 ],
    generators := 
    [ GroupHomomorphismByImages( im, im, [ a1, a2, a3 ], [ a2, a1, a3 ] ),
      GroupHomomorphismByImages( im, im, [ a1, a2, a3 ], [ a1*a2, a2, a3
         ] ) ] ) 

(The function IsIsomorphic tests for isomorphism of p-groups.)

Note: If the component P2.isCompatiblePCentralSeries is not bound it is computed using IsCompatiblePCentralSeries.

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