# A complete classification of finite p-groups all of whose noncyclic subgroups are normal

Abstract: We give a complete classification of finite p-groups all of whose noncyclic subgroups are normal, which solves a problem stated by Berkovich.

## Summary (1 min read)

### 1. Introduction and preliminary results

- The authors consider here only finite p-groups and their notation is standard.
- Let G be a finite non-Dedekindian p-group all of whose noncyclic subgroups are normal.
- But l must normalize 〈f〉 and l 2 inverts 〈f〉, a contradiction.

### 2. Proof of Theorem 1.1

- The authors have obtained the group stated in part (v) of their theorem.
- Now suppose that there exist no elements of order p in G−N and consider the abelian group G/W . (ii) Assume that Z(G) is noncyclic.
- Since Ω1(G) = W ∼= Ep2 , Proposition 1.3 shows that G is also metacyclic and the authors have obtained the groups from part (i) of their theorem.
- It is an easy exercise to show that all the groups from Theorem 1.1 satisfy the assumptions of this theorem and the authors are done.

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### Cites result from "A complete classification of finite..."

...The authors thank Professors Z. Bozikov and Z. Janko for their reference [8]....

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...In particular, we extend the results of Passman, Bozikov, and Janko to non-nilpotent finite groups....

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...The following result was proved by Passman [20] and Bozikov and Janko [8]....

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### "A complete classification of finite..." refers background in this paper

...If G/〈x〉 is noncyclic, then G/〈x〉 is abelian of type (2, 2), n ≥ 1 (since we may assume |G| ≥ 2(4)), and so there is y ∈ G− (〈x〉 ×Z1) such that y(2) ∈ 〈x〉....

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...(v) G = 〈a, b | a(8) = b(8) = 1, a = a, a(4) = b(4)〉, where |G| = 2(5), G ∼= C4, Z(G) ∼= C4, G ′ ∩ Z(G) ∼= C2 and Ω2(G) is abelian of type (4, 2)....

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...We have proved that S is abelian of type (4, 2)....

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...Set S/H = Φ(N/H) so that S is abelian of type (4, 2) because |N : CN (H)| ≤ 2 and H is maximal cyclic....

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...Set T/W = Φ(Q/W ) so that T is abelian of type (4, 2)....

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