Normal Fitting classes and Hall subgroups

Abstract

It was shown by Bryce and Cossey that each Hall π-subgroup of a group in the smallest normal Fitting class S * necessarily lies in S * , for each set of primes π. We prove here that for each set of primes π such that |π| ≥ 2 and π′ is not empty, there exists a normal Fitting class without this closure property. A characterisation is obtained of all normal Fitting classes which do have this property. Let F be a normal Fitting class closed under taking Hall π-subgroups, in the sense of the paragraph above, and let S π denote the Fitting class of all finite soluble π-groups, for some set of primes π. The second main theorem is a characterisation of the groups in the smallest Fitting class containing F and S π in terms of their Hall π-subgroups.