Let G be a finite p-group. We show that if Omega(2)(G) is an extraspecial group then Omega(2)(G) = G (Theorem 1). If we assume only that Q(2)*(G) (the subgroup generated by elements of order p(2)) is an extraspecial group, then the situation is more complicated. If p = 2, then either Omega(2)*(G) = G or G is a semidihedral group of order 16 (Theorem 2). If p > 2, then we can only show that Omega(2)*(G) = Hp(G) (Theorem 3). This is a continuation of our paper [4]. Here we also try to extend our results to finite p-groups with p > 2. Our notation is standard. In particular, if G is a finite p-group and n >= 1 a fixed natural number, then Omega(n) (G) = , Omega(n)*(G) = , (where o(x) denotes the order of an element x), and the Hughes subgroup is defined by H-p(G) = . Also, E-pn denotes the elementary abelian group of order p(n), and M-p3 for p > 2 the non-abelian group of order p(3) and exponent p(2). We state now three known propositions about p-groups of maximal class which are used in this paper.
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