Cover-avoid subgroups in finite solvable groups

Abstract

This paper is concerned with two classes of subgroups of finite solvable groups, S-normalizers, introduced by R. Carter and T. 0. Hawkes in [I], and g-prefrattini subgroups, investigated by T. 0. Hawkes in [2]. Both g-normalizers and F-prefrattini subgroups of a solvable group G cover certain chief factors of G and avoid the remaining chief factors. In this article we are concerned with the following question. To what extent does the coveravoid property characterize the given classes of supgroups in an arbitrary group G ? A permutability condition is given, which together with the coveravoid property does characterize the subgroups in question. T. 0. Hawkes has given an example in [4] which shows the cover-avoid property alone does not characterize F-normalizers in general. An example is given here which shows the cover-avoid property alone does not characterize the prefrattini subgroups. This answers a question raised by W. Gaschiitz in [2]. All groups considered are finite and solvgble. Notation and terminology not explicitly defined is standard.