A note on the index complex of a maximal subgroup

Abstract

Clearly a group G is solvable if all of its chief factors are abelian, in fact G is solvable if each of its maximal subgroups avoids at least one abelian chief factor. In [4] it was announced that G is solvable if each of its maximal subgroups M avoids an abelian section derived from a maximal element of the index complex of M. Also announced was a characterization of the maximal normal solvable subgroup of a group as the intersection of certain maximal subgroups of the group. In this note proofs of these results are presented with some related results. Only finite groups are treated. I. Statements of definitions and results. Let M be a maximal subgroup of group G, M < G. A subgroup C of G is said to be a completion of M in G if C is not contained in M while every proper subgroup of C which is normal in G, is contained in M. The set, I(M), of all completions of M is called the index complex of M in G. The second restriction on a completion C insures that the product of all normal subgroups of G which are proper subgroups of C is itself a proper subgroup of C. It is convenient to define the strict core of a subgroup H + I of G to be the product of all normal subgroups of G which are proper subgroups of H; the strict core of H is denoted by k(H) = kG(H ). Clearly k(H) is proper in H when H~ G but k(H) can differ from H even when H is normal in G, for example, when H is a normal cyclic p-group. (So the strict core of a subgroup can differ from the core.) It follows then that subgroup C of group G is a completion of maximal subgroup M of G, i.e., C ~ I(M), provided (i) G = (C, M) and (ii) k(C) < M. Proposition. The index complex of a maximal subgroup M of group G is nonempty. In particular I (M) contains a normal subgroup of G. Clearly the collection of normal subgroups of G which do not lie in M is nonempty; chose C to be minimal in this partially ordered set. Then CM--(C,M) = G and k(C) < M, so that C ~ Z (M). [] If C is a normal completion of maximal subgroup M of group G, i.e., C< G and C ~ I(M), then C/k(C) is a chief factor of G which is avoided by M. M avoids a chief factor H/K of G if M contains K but not H, in which case H E I (M) and k (H) = K. The set I(M) is partially ordered by set inclusion; maximal elements of I(M) are called