On the upper exponent of a finite \(p\)-group.
For a finite \(p\)-group \(G\) the upper exponent \(\overline\exp(G)\) is \(p^l\) where \(l\) is the length of the series \(G>G^p>(G^p)^p>((G^p)^p)^p>\cdots\). \textit{A. Shalev} [Q. J. Math., Oxf. II. Ser. 42, No. 167, 375-378 (1991; Zbl 0732.20004)] obtained an upper bound for \(\overline\exp(G)\) in terms of the nilpotency class \(c\) and the exponent of \(G\). Answering a question of Y. G. Berkovich the author constructs examples showing that \(\overline\exp(G)\) cannot be bounded in terms of the exponent of \(G\) alone. Actually new upper bounds are obtained that are shown to be attained on certain families of groups of arbitrarily large class. The proof uses the multilinear identities of \textit{M. R. Vaughan-Lee} in associated Lie rings of free Burnside groups of prime-power exponent [The restricted Burnside problem. (Lond. Math. Soc. Monographs. New Series. 8), Oxford: Clarendon Press (1993; Zbl 0817.20001)]. (The author mentions certain gaps in the proof for \(p=2\).)
scientific article; zbMATH DE number 2102482