The subgroups of Baer and Hughes (Q1312368)

For a group \(G\) the Hughes subgroup for exponent \(n\) is defined by \(H_ n(G) = \langle x \in G;\;x^ n \neq 1\rangle\). In connection with Hughes' problem it is known that \(| G : H_ p(G)| = p\) for finite \(G\) and \(p\) prime implies \(H_ p(G)\) is nilpotent. The author deals with arbitrary groups and proves that for a prime \(p\) and soluble \(G\) of derived length \(d\) and exponent \(p^ m\) the condition \(H_ p(G) \neq G\) enforces the nilpotence of \(G\) with class bounded by a function of \(m\) and \(d\). As a consequence analogous results are obtained for soluble \(G\) with derived length \(d\) and exponent \(p^ 2\) in which the norm (= Baer's subgroup \(\kappa(G) = \{g \in G;\;U^ g = U\;\forall U \leq G\})\) is not central, and for soluble groups \(G\) of exponent \(p^ 2\) with a non-trivial power automorphism. Examples show that most of the restrictions imposed in these results cannot be weakened in general.