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Hall-closure and products of Fitting classes

Part of: Structure and classification of infinite or finite groups Representation theory of groups

Published online by Cambridge University Press:  09 April 2009

Owen J. Brison
Owen J. Brison
Affiliation:
Secção de Matemática Pura, Faculdade de Ciências, Avenida 24 de Julho, 134, 3°., 1.300 Lisboa, Portugal
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Abstract

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The Fitting class (of finite, soluble, groups), , is said to be Hall π-closed (where π is a set of primes) if whenever G is a group in and H is a Hall π-subgroup of G, then H belongs to . In this paper, we study the Hall π-closure of products of Fitting classes. Our main result is a characterisation of the Hall π-closedFitting classes of the form (where denotes the so-called smallest normal Fitting class), subject to a restriction connecting π with the characteristic of . We also characterise those Fitting classes (respectively, ) such that (respectively, ) is Hall π-closed for all Fitting classes . In each case, part of the proof uses a concrete group construction. As a bonus, one of these construction also yields a “cancellation result” for certain products of Fitting classes.

MSC classification

Secondary: 20D10: Solvable groups, theory of formations, Schunck classes, Fitting classes, $pi$-length, ranks 20D25: Special subgroups (Frattini, Fitting, etc.) 20E22: Extensions, wreath products, and other compositions
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 32 , Issue 2 , April 1982 , pp. 145 - 164
Copyright
Copyright © Australian Mathematical Society 1982

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