FINITE p-GROUPS WHICH ARE NOT GENERATED BY THEIR NON-NORMAL SUBGROUPS

Here we classify finite non-Dedekindian p-groups which are not generated by their non-normal subgroups. (Theorem 1). The purpose of this paper is to classify non-Dedekindian finite p-groups which are not generated by their non-normal subgroups. It is surprising that such p-groups must be of class 2 with a cyclic commutator subgroup. We consider here only finite p-groups and our notation is standard (see [1]). We prove the following result. Theorem 1. Let G be a non-Dedekindian p-group and let G0 be the subgroup generated by all nonnormal subgroups of G, where we assume G0 < G. Then G is of class 2, G/G0 is cyclic and for each g ∈ G − G0, {1} 6 = 〈g〉 ∩G0 EG and G/(〈g〉 ∩G0) is abelian so that G ′ is cyclic. Proof. Since our group G has at least p (non-normal) conjugate cyclic subgroups, it follows that the subgroup G0 is noncyclic. Let x ∈ G−G0. Then 〈x〉EG, by hypothesis, and so G centralizes 〈x〉. It follows from 〈G−G0〉 = G that G ≤ Z(G) and so cl(G) = 2. Let g ∈ G −G0. Then Z = 〈g〉 ⊳ G. Write Z0 = Z ∩ G0; then Z0, being the intersection of two G-invariant subgroups, is G-invariant. We claim that G/Z0 is Dedekindian. Indeed, let X/Z0 be any proper subgroup in G/Z0. We have to show that X ⊳ G. If X 6≤ G0, then X E G. Now assume that X < G0 (the subgroup G0 is G-invariant). Then XZ = ZX is normal in G 2010 Mathematics Subject Classification. 20D15.