SOME PECULIAR MINIMAL SITUATIONS BY FINITE p-GROUPS
نویسندگان
چکیده
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 = 〈a, b | a n−1 = b = 1, n ≥ 3, a = a1+p n−2 〉, 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 H of each nonnormal subgroup H of G is the largest possible, i.e., |G : H| = 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
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