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- W2054409620 abstract "During his investigation of the possible non-Hopf kernels for finitely generated groups in [1], Dey proves that the minimum number of generators d ( G n ) of the n -th direct power G n 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 ( G n )} 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." @default.
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- W2054409620 date "1974-03-01" @default.
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- W2054409620 title "Growth sequences of finite groups" @default.
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- W2054409620 doi "https://doi.org/10.1017/s1446788700016712" @default.
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