On the Hughes conjecture for some finite p-groups

Let G be a group, p a prime and \(H_p(G)\) $H_p(G)$ the subgroup of G generated by the elements of order different from p. In 1957, D. R. Hughes conjectured that either \(H_p(G)=1\) $H_p(G)=1$ , \(H_p(G)=G\) $H_p(G)=G$ , or \([G:H_p(G)]=p\) $[G:H_p(G)]=p$ . In this paper, we prove this conjecture for finite extraspecial p-groups (where \(p>2\) $p>2$ ), 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.