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The Schur-Zassenhaus theorem in locally finite groups

Published online by Cambridge University Press:  09 April 2009

B. Hartley
B. Hartley
Affiliation:
Mathematics Institute, University of Warwick, Coventry, CV4 7AL, England.
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Abstract

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Let L = HK be a semidirect product of a normal locally finite π′-group H by a locally finite π′-group K, where π, is a set of primes. Suppose CK(H) = 1 and L is Sylow π-sparse (which in the countable case just says that the Sylow π-subgroups of L are conjugate). This paper completes the characterization of those groups which can occur as K—this had previously been obtained under the assumption that L is locally soluble. The answer is the same—essentially that the groups occurring are those having a subgroup of finite index which is a subdirect product of so-called “pinched” groups.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 22 , Issue 4 , December 1976 , pp. 491 - 493
Copyright
Copyright © Australian Mathematical Society 1976

References

Gorenstein, D (1968) Finite groups (Harper and Row, New York, 1968).Google Scholar
Hall, P. (1956) ‘Theorems like Sylow's’, Proc. London Math. Soc. (3) 6, 286–304.CrossRefGoogle Scholar
Hartley, B. (1971) ‘Sylow subgroups of locally finite groups’, Proc. London Math. Soc. (3) 23, 159–192.CrossRefGoogle Scholar
Hartley, B. (1972) ‘Sylow theory in locally finite groups’, Compositio Math. 25, 263–280.Google Scholar
Hartley, B. (1975) ‘A class of modules over a locally finite group II’, J. Austral. Math. Soc. Ser A 19, 437–469.CrossRefGoogle Scholar
Kargapolov, M. I. (1959) ‘Some problems in the theory of nilpotent and soluble groups’, Dokl. Akad. Nauk SSSR, 127 1164–1166 (Russian).Google Scholar
Rae, A. (1972) ‘Local systems and Sylow subgroups in locally finite groups’, Proc. Cambridge Philos. Soc. 75, 1–22.CrossRefGoogle Scholar
Sunkov, V. P. (1971) ‘On locally finite groups of finite rank’, Algebra i Logika 10, 199–225 (Russian). Algebra and Logic 10 (1971), 127–142.Google Scholar