7.43 SubnormalSeries

SubnormalSeries( G, U )

Let U be a subgroup of G, then SubnormalSeries returns a subnormal series <G> = G_1 > ... > G_n of groups such that U is contained in G_n and there exists no proper subgroup V between G_n and U which is normal in G_n.

G_n is equal to U if and only if U is subnormal in G.

Note that this function may not terminate if G is an infinite group.

    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> SubnormalSeries( s4, c2 );
    [ Group( (1,2,3,4), (1,2) ) ]
    gap> IsSubnormal( s4, c2 );
    false
    gap> c2 := Subgroup( s4, [ (1,2)(3,4) ] );
    Subgroup( Group( (1,2,3,4), (1,2) ), [ (1,2)(3,4) ] )
    gap> SubnormalSeries( s4, c2 );
    [ Group( (1,2,3,4), (1,2) ), Subgroup( Group( (1,2,3,4), (1,2) ), 
        [ (1,2)(3,4), (1,3)(2,4) ] ),
      Subgroup( Group( (1,2,3,4), (1,2) ), [ (1,2)(3,4) ] ) ]
    gap> IsSubnormal( s4, c2 );
    true 

The default function GroupOps.SubnormalSeries constructs the subnormal series as follows. G_1 = G and G_{i+1} is set to the normal closure (see NormalClosure) of U under G_i. The length of the series is n, where n = max{i; G_i > G_{i+1}}.

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