7.63 IsSubnormal

IsSubnormal( G, U )

IsSubnormal returns true if the subgroup U of G is subnormal in G and false otherwise.

A subgroup U of G is subnormal if and only if there exists a series of subgroups <G> = G_0 > G_1 > ... > G_n = <U> such that G_i is normal in G_{i-1} for all i in {1, ..., n}.

Note that U must be a subgroup of G. The function sets and checks U.isSubnormal if G is the parent group of G.

    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> 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> IsSubnormal( s4, c2 );
    true 

The default function GroupOps.IsSubnormal uses SubnormalSeries (see SubnormalSeries) in order to check if U is subnormal in G.

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