Elements of order at most 4 in finite 2-groups, 2

Abstract

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 Omega*_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)=H_p(G)(Theorem 3).