Some peculiar minimal situations by finite p-groups

In this paper we show that a finite p-group which possesses non-normal subgroups and such that any two non-normal subgroups of the same order are conjugate must be isomorphic to Mpn = , where in case p = 2 we must have n ≥ 4. This solves Problem Nr. 1261 stated by Y. Berkovich in [1]. In a similar way we solve Problem Nr. 1582 from [1] by showing that Mpn is the only finite p-group with exactly one conjugate class of non-normal cyclic subgroups. Then we determine up to isomorphism all finite p-groups which possess non-normal subgroups and such that the normal closure HG of each non-normal subgroup H of G is the largest possible, i.e., |G : HG| = p. It turns out that G is either the nonabelian group of order p3, p > 2, and exponent p or G is metacyclic. This solves the Problem Nr. 1164 stated by Berkovich [1]. We classify also finite 2-groups with exactly two conjugate classes of four-subgroups. As a result, we get three classes of such 2-groups. This solves Problem Nr. 1260 stated by Y.Berkovich in [1]