GAP Manual: 7.27. Normalizer

Normalizer( S, U )

Let S and U be groups with a common parent group G. Then Normalizer returns the normalizer of U in S.

The normalizer N_(S)( U ) of U in S is the biggest subgroup of S which leaves U invariant under conjugation.

If S is the parent group of U then Normalizer sets and tests U.normalizer.

    gap> s4 := Group( (1,2,3,4), (1,2) );
    Group( (1,2,3,4), (1,2) )
    gap> c2 := Subgroup( s4, [ (1,2) ] );
    Subgroup( Group( (1,2,3,4), (1,2) ), [ (1,2) ] )
    gap> Normalizer( s4, c2 );
    Subgroup( Group( (1,2,3,4), (1,2) ), [ (3,4), (1,2) ] ) 

The default function GroupOps.Normalizer uses Stabilizer (see Stabilizer) in order to compute the stabilizer of U in S acting by conjugation (see ConjugateSubgroup).