On theta pairs for a maximal subgroup

For a maximal eubgroup M of a finite group G, a 8-pair is any pair of subgroups (C,D) of G such that (i) D?G, D≤C, (ii) - G, - M and (iii) C/D has no proper normal subgroup of G/D. A partial order may be defined on the family of 8-pairs. Let △(M) - {(C,D)|(C,D) is a maximal 8-pair and CM - G}. The purpose of this note is to prove: (1) A group G is solvable if and only if, for each maximal subgroup M of G, △(M) contains a 8-pair (C,D) such that C/D ie nilpctent. (2) If a group G is S₄-free, then G ia eupersolvable if and only if, for each maximal subgroup M of G, △(M) contains a 8-pair (C,D) auch that C/D is cyclic