A complete classification of finite p-groups all of whose noncyclic subgroups are normal
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.
Did you find this useful? Give us your feedback
Citations
References
46 citations
"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〉....
[...]
...(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)....
[...]
...We have proved that S is abelian of type (4, 2)....
[...]
...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....
[...]
...Set T/W = Φ(Q/W ) so that T is abelian of type (4, 2)....
[...]
42 citations
19 citations
"A complete classification of finite..." refers background in this paper
...(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)....
[...]
...We have proved that S is abelian of type (4, 2)....
[...]
...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....
[...]
...Set T/W = Φ(Q/W ) so that T is abelian of type (4, 2)....
[...]
...Set Z0 = M ∩ Z, where |Z0 : W | = 2 and Z0 is abelian of type (4, 2)....
[...]