On the Hughes conjecture for some finite p-groups

Abstract

Let G be a group, p a prime and \(H_p(G)\) the subgroup of G generated by the elements of order different from p. In 1957, D. R. Hughes conjectured that either \(H_p(G)=1\), \(H_p(G)=G\), or \([G:H_p(G)]=p\). In this paper, we prove this conjecture for finite extraspecial p-groups (where \(p>2\)), finite minimal non-abelian p-groups and finite non-abelian p-groups having cyclic maximal subgroup. Moreover, we give some sufficient conditions for 2-generated finite non-abelian p-groups which guarantee the existence of the Hughes conjecture.