On a~connection between Hughes' conjecture and relations in finite groups of prime exponent

E. I. Khukhro

Sbornik. Mathematics, Tome 44 (1983) no. 2, pp. 227-237

Résumé

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.