25.91 CoprimeComplement

CoprimeComplement( U, N )

CoprimeComplement returns a complement of a normal p-elementary abelian Hall subgroup N of U.

Note that, as N is a normal Hall-subgroup of U, the theorem of Schur guarantees the existence of a complement.

    gap> s4xc25 := DirectProduct( s4, CyclicGroup( AgWords, 25 ) );
    Group( a1, a2, a3, a4, b1, b2 )
    gap> s4xc25.name := "s4xc25";;
    gap> a4xc25 := Subgroup( s4xc25,
    >                  Sublist( s4xc25.generators, [2..5] ) );
    Subgroup( s4xc25, [ a2, a3, a4, b1 ] )
    gap> N := Subgroup( s4xc25, [ s4xc25.3, s4xc25.4 ] );
    Subgroup( s4xc25, [ a3, a4 ] )
    gap> CoprimeComplement( a4xc25, N );
    Subgroup( s4xc25, [ a2, b1, b2 ] ) 

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