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We determine up to isomorphism finite non-Dedekindian p-groups G (i.e., p-groups which possess non-normal subgroups) such that the normal closure of each non-normal cyclic subgroup in G is nonabelian. It turns out that we must have p = 2 and G has an abelian maximal subgroup A of exponent 2e, e ≥ 3, and an element v ∈ G−A such that for all h ∈ A we have either hv = h−1 or hv = h−1+2 e−1 . Let G be a finite p-group. Then it is well known (see [3, Theorem 224.1]) that the normal closure of each cyclic subgroup in G is abelian if and only if each two-generator subgroup of G is of class ≤ 2. If each two-generator subgroup of a p-group G is of class ≤ 2, then either G is of class ≤ 2 or p = 3 and G is of class 3. It is natural to ask what happens if G is a non-Dedekindian finite p-group in which the normal closure of each non-normal cyclic subgroup is nonabelian. It turns out as a big surprise that in this case we must have p = 2 and G can be determined up to isomorphism (Theorem 2). All groups considered here are finite p-groups and our notation is standard (see [1]). Definition 1. Let M be a 2-group possessing an abelian maximal subgroup H of exponent ≥ 4 such that there is an element v ∈ M −H which inverts each element of H. Then o(v) ≤ 4 since v inverts 〈v〉 ≤ H. If o(v) = 2, then all elements in M −H are involutions and M is called ”quasidihedral” (or generalized dihedral). 2010 Mathematics Subject Classification. 20D15.

Finite p-groups in which the normal closure of each non-normal cyclic subgroup is nonabelian