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Journal ArticleDOI

Finite metahamiltonian p-groups

15 Nov 2015-Journal of Algebra (Academic Press)-Vol. 442, Iss: 442, pp 23-35
TL;DR: In this paper, some properties of finite metahamiltonian p-groups are investigated and these properties are used in classifying metahammiltonian P-groups, which is a natural generalization of Hamiltonian groups.
About: This article is published in Journal of Algebra.The article was published on 2015-11-15 and is currently open access. It has received 12 citations till now.
Citations
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Journal ArticleDOI
TL;DR: In this article, the authors give a criterion for finite groups whose nonnormal subgroups are metacyclic, and based on the criterion, the $p$-groups whose non-normal subgroup are metaceclicare classified up to isomorphism.
Abstract: For an odd prime $p$, we give a criterion for finite$p$-groups whose nonnormal subgroups are metacyclic, and basedon the criterion, the $p$-groups whose nonnormal subgroups are metacyclicare classified up to isomorphism. Thissolves a problem proposed by Berkovich.

3 citations

Journal ArticleDOI
TL;DR: In this paper, the maximum and minimum order of the non-normal subgroups of a finite p-group is defined, and groups G such that M(G) < 2m (G) − 1.
Abstract: Suppose that G is a finite p-group. If G is not a Dedekind group, then G has a non-normal subgroup. We use pM(G) and pm(G) to denote the maximum and minimum of the orders of the non-normal subgroups of G; respectively. In this paper, we classify groups G such that M(G) < 2m(G)‒1: As a by-product, we also classify p-groups whose orders of non-normal subgroups are pk and pk+1.

3 citations

Journal ArticleDOI
01 Jun 2021
TL;DR: In this paper, a finite metahamiltonian p-group is classified up to isomorphism, and the structure of the p-groups is determined by the Sylow subgroups.
Abstract: A finite non-abelian group G is called metahamiltonian if every subgroup of G is either abelian or normal in G. If G is non-nilpotent, then the structure of G has been determined. If G is nilpotent, then the structure of G is determined by the structure of its Sylow subgroups. However, the classification of finite metahamiltonian p-groups is an unsolved problem. In this paper, finite metahamiltonian p-groups are completely classified up to isomorphism.

3 citations

Journal ArticleDOI
TL;DR: In this article , the authors study soluble groups in which every subgroup is abelian or pronormal, and they extend the well-known class of metahamiltonian groups.
Abstract: A subgroup H of a group G is said to be pronormal in G if each of its conjugates Hg in G is already conjugate to it in the subgroup ⟨H,Hg⟩. Extending the well-known class of metahamiltonian groups, we study soluble groups in which every subgroup is abelian or pronormal.

2 citations

References
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BookDOI
01 Jan 1967

5,518 citations

Journal ArticleDOI
TL;DR: In this article, the finite p -groups all of whose non-abelian proper subgroups are generated by two elements are classified and a classification of finite p-groups with two elements is given.

64 citations

Journal ArticleDOI
01 Jul 1958
TL;DR: In this article, the Frattini subgroup of a group of order a power of the prime p was derived from the derived group of G and the lower central series of G was written.
Abstract: Let G denote a group of order a power of the prime p, and let G′ be the derived group of G. The lower central series of G will be writtenFor any subgroup H of G we denote by P(H) the subgroup of H generated by all elements xp as x runs through H, and by Φ(H) the Frattini subgroup of H. We write (H:Φ(H)) = pd(H); thus d(H) is the minimal number of generators of H.

59 citations