7.29 PrefrattiniSubgroup

PrefrattiniSubgroup( G )

PrefrattiniSubgroup returns a Prefrattini subgroup of the group G.

A factor M/N of G is called a Frattini factor if M/N leq phi(G/N) holds. The group P is a Prefrattini subgroup of G if P covers each Frattini chief factor of G, and if for each maximal subgroup of G there exists a conjugate maximal subgroup, which contains P.

    gap> s4 := SymmetricGroup( AgWords, 4 );;
    gap> ss4 := SpecialAgGroup( s4 );;
    gap> PrefrattiniSubgroup( ss4 );
    Subgroup( Group( g1, g2, g3, g4 ), [  ] ) 

Currently PrefrattiniSubgroup can only be applied to special Ag groups (see Special Ag Groups).

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