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On a connection between Hughes' conjecture and relations in finite groups of prime exponent
E. I. Khukhro
Abstract:
Under the assumption that the ideal of relations of a free 3-generator group of period
$p$ does not coincide modulo
$p$ with the
$(p-1)$-Engel ideal it is proved that there exist
$p$-groups
$P$ of nilpotence degree
$2p-1$ in which the index of the Hughes subgroup
$H_p(P)$ is
$p^2$ (Theorem 1). The author also finds that Macdonald's result on
$p$-groups of class
$2p-2$ is best possible (at least for
$p=5,7,11$). The proof is based on direct computations almost the same as in work of A. I. Kostrikin dating from 1957; it uses properties of the coefficients in the Baker–Hausdorff formula.
An automorphism
$\varphi$ of order
$p$ of the group
$G$ is called splitting if $xx^\varphi x^{\varphi\,2}\dots x^{\varphi\,p-1}=1$ for all
$x$ in
$G$. It is easy to see that
$G\ne H_p(G)$ if and only if
$G=G_1\langle\varphi\rangle$, where
$\varphi$ is a splitting automorphism of order
$p$ of
$G_1$. It is proved that if a finite
$p$-group
$P$ admits a splitting automorphism
$\varphi$ of order
$p$ and the nilpotency degree of
$P\langle\varphi\rangle$ does not exceed
$2p-2$, then
$P$ is regular (Theorem 2). From Theorem 2 it is possible to deduce an independent proof of Hughes' conjecture for
$p$-groups of class
$2p-2$.
On the basis of Theorem 1 the author constructs examples of
$p$-groups admitting a splitting automorphism of order
$p$ for which the associated Lie ring is not a
$(p-1)$-Engel ring.
Bibliography: 12 titles.
UDC:
519.4
MSC: Primary
20D15; Secondary
20F40,
20F45 Received: 03.10.1980