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Growth sequences of finite groups

Published online by Cambridge University Press:  09 April 2009

James Wiegold
James Wiegold
Affiliation:
University College, Cardiff, U. K.
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During his investigation of the possible non-Hopf kernels for finitely generated groups in [1], Dey proves that the minimum number of generators d(Gn) of the n-th direct power Gn of a non-trival finite group G tends to infinity with n. This has prompted me to ask the question: what are the ways in which the sequence {d(Gn)} can tend to infinity? Let us call this the growth sequence for G; it is evidently monotone non-decreasing, and is at least logarithmic (Theorem 2.1). This paper is devoted to a proof that, broadly speaking, there are two different types of behaviour. If G has non-trivial abelian images (the imperfect case, § 3), then the growth sequence of G is eventually an arithmetic progression with common difference d(G/G'). In special cases (Theorem 5.2) the initial behaviour can be quite nasty. Our arguments in § 3 are totally elementary. If G has only trivial abelian images (the perfect case,§ 4), then the growth sequence of G is eventually bounded above by a sequence that grows logarithmically. It is a simple consequence of this fact that there are arbitrarily long blocks of positive integers on which the growth sequence takes constant values. This is a characteristic property of perfect groups, and indeed it was this feature in the growth sequences of large alternating groups (which I found by using ad hoc permutational arguments) that attracted me to the problem in the first place. The discussion of the perfect case rests on the lovely paper of Hall [2], which was brought to my notice by M. D. Atkinson.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 17 , Issue 2 , March 1974 , pp. 133 - 141
Copyright
Copyright © Australian Mathematical Society 1974

References

[1]Dey, I. M. S., ‘Embeddings in non-Hopf groups’, J. London Math. Soc. (2) 1 (1969), 745–749.CrossRefGoogle Scholar
[2]Hall, P., ‘The Eulerian functions of a group’, Quart. J. Math. (Oxford) 7 (1936), 134–151.Google Scholar
[3]Neumann, B. H., ‘Identical relations in groups I’, Math. Ann. 114 (1937), 506–525.CrossRefGoogle Scholar
[4]Neumann, Hanna, Varieties of groups (Egebnisse der Mathematik and ihrer Grenzgebiete, 37, 1967).CrossRefGoogle Scholar
[5]Tyrer-Jones, J. M.. ‘Direct products and the Hopf property’, J. Austral. Math. Soc. 17 (1974), 174–196.CrossRefGoogle Scholar