7.25 NormalClosure

NormalClosure( S, U )

Let S and U be groups with a common parent group G. Then NormalClosure returns the normal closure of U under S as a subgroup of G.

The normal closure N of a group U under the action of a group S is the smallest subgroup in G that contains U and is invariant under conjugation by elements of S. Note that N is independent of G.

    gap> s4 := Group( (1,2,3,4), (1,2) );
    Group( (1,2,3,4), (1,2) )
    gap> s4.name := "s4";;
    gap> d8 := Subgroup( s4, [ (1,2,3,4), (1,2)(3,4) ] );
    Subgroup( s4, [ (1,2,3,4), (1,2)(3,4) ] )
    gap> NormalClosure( s4, d8 );
    Subgroup( s4, [ (1,2,3,4), (1,2)(3,4), (1,3,4,2) ] )
    gap> last = s4;
    true 

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