特定阶的子群都同构且交换的有限p-群

研究某些子群同构的有限p-群是很有趣的.例如,Hermann和Mann都曾研究过极大子群都同构的有限p-群,但这类群的结构非常复杂,到现在人们都没能给出其分类.研究了特定阶的子群都同构且交换的有限p-群,并给出其分类.     

Finite non-elementary abelian p-groups whose number of subgroups is maximal

Assume G is a direct product of M _p (1, 1, 1) and an elementary abelian p-group, where M _p (1, 1, 1) = 〈a, b | a ^p = b ^p = c ^p =1, [a,b]=c,[c,a] = [c,b]=1〉. When p is odd, we prove that G is the group whose number of subgroups is maximal except for elementary abelian p-groups. Moreover, the counting formula for the groups is given.     

Finite p-groups all of whose non-abelian proper subgroups are metacyclic

In this paper, we classify the finite p-groups all of whose non-abelian proper subgroups are metacyclic and answer a question posed by Berkovich. Received: 22 June 2005     

Finite nonabelian p-groups all of whose nonabelian maximal subgroups are either metacyclic or minimal nonabelian

We give here a complete classification of the title groups (Theorem A).     

Finite p-groups whose nonnormal subgroups are metacyclic

For an odd prime p, we give a criterion for finite p-groups whose nonnormal subgroups are metacyclic, and based on the criterion, the p-groups whose nonnormal subgroups are metacyclic are classified up to isomorphism. This solves a problem proposed by Berkovich.     

Finite p-groups all of whose proper subgroups have small derived subgroups

Let G be a finite p-group.If the order of the derived subgroup of each proper subgroup of G divides pi,G is called a Di-group.In this paper,we give a characterization of all D1-groups.This is an answer to a question introduced by Berkovich.     

Finite p-groups all of whose non-abelian proper subgroups are generated by two elements

In this paper, we classify the finite p-groups all of whose non-abelian proper subgroups are generated by two elements.     

Finite <Emphasis Type="Italic">p</Emphasis>-groups all of whose maximal abelian subgroups are soft

A subgroup A of a p-group G is said to be soft in G if C _G (A) = A and |N _G (A/A| = p. In this paper we determined finite p-groups all of whose maximal abelian subgroups are soft; see Theorem A and Proposition 2.4.     

Finite p-groups with complemented maximal cyclic subgroups

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 6, pp. 780–785, June, 1992.     

Finite groups all of whose second maximal subgroups are {\mathcal{H }}_p-groups

Let $G$ be a finite group. A subgroup $H$ of $G$ is called an $\mathcal{H }$ -subgroup of $G$ if $N_G(H)\cap H^g\le H$ for all $g\in G$ . A group $G$ is said to be an ${\mathcal{H }}_p$ -group if every cyclic subgroup of $G$ of prime order or order 4 is an $\mathcal{H }$ -subgroup of $G$ . In this paper, the structure of a finite group all of whose second maximal subgroups are ${\mathcal{H }}_p$ -subgroups has been characterized.     

Finite groups with abelian second maximal subgroups

Abstract

In this article, we give a complete classification of finite groups whose second maximal subgroups are all abelian.     

Groups whose proper factors are all abelian

The question implied in the title is a problem of A. Zaks, namely, which finite groups (other than abelian and simple groups) have all their proper factors abelian? This paper answers the question in the case of groups with non-trivial centre, or, equivalently, in the case ofp-groups, and gives a structure theorem for such groups.