7.33 FactorGroup

FactorGroup( G, N )

FactorGroup returns the factor group <G> / <N> where N must be a normal subgroup of G (see IsNormal). This is the same as G / N (see Operations for Groups).

NaturalHomomorphism returns the natural homomorphism from G (or a subgroup thereof) onto the factor group (see NaturalHomomorphism).

It is not specified how the factor group N is represented.

    gap> a4 := Group( (1,2,3), (2,3,4) );;  a4.name := "a4";
    "a4"
    gap> v4 := Subgroup(a4,[(1,2)(3,4),(1,3)(2,4)]);;  v4.name := "v4";
    "v4"
    gap> f := FactorGroup( a4, v4 );
    (a4 / v4)
    gap> Size( f );
    3
    gap> Elements( f );
    [ FactorGroupElement( v4, () ), FactorGroupElement( v4, (2,3,4) ), 
      FactorGroupElement( v4, (2,4,3) ) ] 

If G is the parent group of N, FactorGroup first checks for the knowledge component N.factorGroup. If this component is bound, FactorGroup returns its value. Otherwise, FactorGroup calls G.operations.FactorGroup( G, N ), remembers the returned value in N.factorGroup, and returns it. If G is not the parent group of N, FactorGroup calls G.operations.FactorGroup( G, N ) and returns this value.

The default function called this way is GroupOps.FactorGroup. It returns the factor group as a group of factor group elements (see FactorGroupElement). Look under FactorGroup in the index to see for which groups this function is overlaid.

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