25.90 Complementclasses

Complementclasses( U, N )

Let U and N be ag groups such that N is a normal subgroup of U. Complementclasses returns a list of representatives for the conjugacy classes of complements of N in U.

Note that the empty list is returned if U does not split over N.

Complementclasses descends along an elementary abelian series of U containing N. See CNW90 for details.

    gap> v4 := Subgroup( s4, [ c, d ] );
    Subgroup( s4, [ c, d ] )
    gap> Complementclasses( s4, v4 );
    [ Subgroup( s4, [ a, b ] ) ]
    gap> z4 := CyclicGroup( AgWords, 4 );
    Group( c4_1, c4_2 )
    gap> z2 := Subgroup( z4, [ z4.2 ] );
    Subgroup( Group( c4_1, c4_2 ), [ c4_2 ] )
    gap> Complementclasses( z4, z2 );
    [  ]
    gap> m9 := ElementaryAbelianGroup( AgWords, 9 );
    Group( m9_1, m9_2 )
    gap> m3 := Subgroup( m9, [ m9.2 ] );
    Subgroup( Group( m9_1, m9_2 ), [ m9_2 ] )
    gap> Complementclasses( m9, m3 );
    [ Subgroup( Group( m9_1, m9_2 ), [ m9_1 ] ),
      Subgroup( Group( m9_1, m9_2 ), [ m9_1*m9_2 ] ),
      Subgroup( Group( m9_1, m9_2 ), [ m9_1*m9_2^2 ] ) ] 

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