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 Sπ 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 Sπ in terms of their Hall π-subgroups.
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