ON THETA-PAIRS FOR MAXIMAL-SUBGROUPS

For a maximal subgroup M of a finite group G, a theta-pair is any pair of subgroups (C,D) of G such that (i) D<G, D<C, (ii) (M, C)=G, (M,D)=M and (iii) C/D has no proper normal subgroup of G/D. A natural partial ordering is defined on the family of theta- pairs. We study the further properties of the maximal theta-pairs of M and obtain several results on theta-pairs which imply G to be pi-solvable, pi-supersolvable and pi-nilpotent.