The influence on a finite group of the cofactors and subcofactors of its subgroups

Abstract:

The effect on a finite group G of imposing a condition $\theta$ on its proper subgroups has been studied by Schmidt, Iwasawa, Itô, Huppert, and others. In this paper, the effect on G of imposing $\theta$ on only the cofactor $H/{\text {cor}_G}\;H$ (or more generally, the subcofactor $H/{\text {scor}_G}\;H$) of certain subgroups H of G is investigated, where ${\text {cor}_G}\;H\;({\text {scor}_G}\;H)$ is the largest G-normal (G-subnormal) subgroup of H. It is shown, for example, that if (a) $H/{\text {scor}_G}\;H$ is p-nilpotent for all self-normalizing $H < G$, or if (b) $H/{\text {scor}_G}\;H$ is p-nilpotent for all abnormal $H < G$ and p is odd or the p-Sylows of G are abelian, then in either case, G has a normal p-subgroup P for which G/P is p-nilpotent. Results of this type are also derived for $\theta =$ nilpotent, nilpotent of class $\leqq n$, solvable of derived length $\leqq n,\sigma$-Sylow-towered, supersolvable. In some cases, additional structure in G is obtained by imposing $\theta$ not only on these “worst” parts of the “bad” subgroups of G (from the viewpoint of normality), but also on the “good” subgroups, those which are normal in G or are close to being normal in that their cofactors are small. Finally, this approach is in a sense dualized by an investigation of the influence on G of the outer cofactors of its subgroups. The consideration of nonnormal outer cofactors is reduced to that of the usual cofactors. The study of normal outer cofactors includes the notion of normal index of maximal subgroups, and it is proved, for example, that G is p-solvable iff the normal index of each abnormal maximal subgroup of G is a power of p or is prime to p.