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On soluble groups of prime-power exponent

Published online by Cambridge University Press:  17 April 2009

Warren Brisley
Affiliation:
University of Newcastle, Newcastle, New South Wales
L.G. Kovács
Affiliation:
Institute of Advanced Studies, Australian National University, Canberra, ACT.
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Abstract

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Let p be a prime and the variety of elementary abelian by elementary abelian p-groups. A result of Brisley and Macdonald is generalized as follows. If H is a finite group in and G is a soluble group of p–power exponent such that no section of G is isomorphic to H, then G is nilpotent and its class is bounded by a function of three variables: H, the exponent of G, and the soluble length of G. It is a corollary that if the variety generated by a soluble group G of finite exponent contains , then each finite group in is isomorphic to some section of G.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1971

References

[1]Brisley, Warren and Macdonald, I. D., “Two classes of metabelian p-groups”, Math. Z. 112 (1969), 512.CrossRefGoogle Scholar
[2]Gupta, N. D. and Newman, M. F., “On metabelian groups”, J. Austral. Math. Soc. 6 (1966), 362368.CrossRefGoogle Scholar
[3]Hall, P., “Some sufficient conditions for a group to be nilpotent”, Illinois J. Math. 2 (1958), 787801.CrossRefGoogle Scholar
[4]Higman, Graham, “Some remarks on varieties of groups”, Quart. J. Math. Oxford (2) 10 (1959), 105178.CrossRefGoogle Scholar
[5]Neumann, Hanna, Varieties of groups (Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 37, Springer-Verlag, Berlin, Heidelberg, New York, 1967).CrossRefGoogle Scholar
[6]Stewart, A. G. R., “On the class of certain nilpotent groups”, Proc. Roy. Soc. Ser. A 292 (1966), 374379.Google Scholar