On finite \(p\)-groups not satisfying the Hughes conjecture
The title refers to those finite \(p\)-groups for which the Hughes subgroup \(H_p(G):=\langle x\in G\mid x^p\neq 1\rangle\) is nontrivial and has index \(p^k>p\) in \(G\). Here, \(G\) stands for such a group. Let \(d\), \(m\) denote the minimal numbers of generators of \(G\), \(G/H_p(G)\) respectively and let \(c\), \(e\) be their respective nilpotency classes. It is shown (Theorem 2) that, if a multilinear identity of degree \(\leq(k- 1)(p-1)+1\) holds in the associated Lie ring of every finite group of exponent \(p\), then it also holds in the associated Lie ring of \(G\). It follows (Corollary 1) that \(c\leq k(d,p)\), where \(k(d,p)\) is the Kostrikin bound to the nilpotency class of a \(d\)-generator \((p-1)\)-Engel Lie ring of characteristic \(p\). It is also shown that, for each fixed \(p\), \(k(d,p)\) is bounded above by a linear function of \(d\). The following Theorem 4 (due to E. I. Zel'manov and the author) is deduced from Theorem 3 by a simple but ingenious argument: if the conjecture is true for a given \(p\), then the class \(e\) of \(G/H_p(G)\) is bounded for \(p\)-groups \(G\). With some further argument, it is deduced that \(G/H_p (G)\) must be abelian when \(p=5\).
scientific article; zbMATH DE number 794630