Two Important Classes of p-groups and the Orders of Their Automorphism Groups

In the paper we obtain two infinite classes of p-groups, calculate the orders of their automorphism groups and correct a mistake（perhaps misprinted） of Rodney James＇ paper in 1980.

Factorization Numbers of a Class of Finite p-groups

Let p be a prime number and f_2(G) be the number of factorizations G = AB of the group G, where A, B are subgroups of G. Let G be a class of finite p-groups as follows,G = a, b | a^(p^n)= b^(p^m)= 1, a^b= a^(p^(n-1)+1), where n > m ≥ 1. In this article, the factorization number f_2(G) of G is computed, improving the results of Saeedi and Farrokhi in [5].

On the Structure of the Augmentation Quotient Group for Some Non-abelian p-groups

In this paper, we study the basis of augmentation ideals and the quotient groups of finite non-abelian p-group which has a cyclic subgroup of index p, where p is an odd prime, and k is greater than or equal to 3. A concrete basis for the augmentation ideal is obtained and then the structure of its quotient groups can be determined.

FINITE p-GROUPS WHICH CONTAIN A SELF-CENTRALIZING CYCLIC NORMAL SUBGROUP

For any prime p, all finite noncyclic p-groups which contain a self-centralizing cyclic normal subgroup are determined by using cohomological techniques. Some applications are given, including a character theoretic description for such groups.

On the Factorization Numbers of a Class of Finite p-Groups

Let G be a group and A and B be subgroups of G.If G=AB,then G is said to be factorized by A and B.Let p be a prime number.The factorization numbers of a 2-generators abelian p-group and a modular p-group have been determined.Further,suppose that G is a finite p-group as follows G=<a,b|a^(p)^(n)=b^(p)^(m)=1,a^(b)=a^(p^(n-1)+1)>,where n≥2,m≥1.In this paper,the factorization number of G is computed completely,which is a generalization of the result of Saeedi and Farrokhi.

Finite p-groups with abelian maximal subgroups generated by two elements

Assume that G is a finite non-abelian p-group.If G has an abelian maximal subgroup whose number of Generators is at least n,then G is called an M_(n)-group.For p=2,M_(2)-groups have been classified.For odd prime p,this paper provides the isomorphism classification of M_(2)-groups,thereby achieving a complete classification of M_(2)-groups.

关于内Cernikov群

设群G的任意真子群都是Cernikov群,但它本身不是Cernikov群,那么称G是内Cernikov群(MNC-群).研究内Cernikov群的一些性质,并且给出内Cernikov群的例子.同时,利用内Cernikov群的性质得到内FCmin-群的有趣的结果.

The Automorphism Group of a p-Group of Order p^n with an Element of Order p^(n-2)

A finite p-group G is called an LA-group if |G|||Aut(G)| when G is non-cyclic and |G|>p^2. This paper shows that a p-group of order p^n with an element of order p^(n-2) is an LA-group.

一类无限Cernikov p-群的自同构群

设G是无限Cernikov p-群,且G的每个真商群是Abel群,但G不是Abel群,本文确定了G的自同构群.

On Non-commuting Sets in a Finite p-group with Derived Subgroup of Prime Order

Let G be a finite group. A nonempty subset X of G is said to be noncommuting if xy≠yx for any x, y ∈ X with x≠y. If ｜X｜ ≥ ｜Y｜ for any other non-commuting set Y in G, then X is said to be a maximal non-commuting set. In this paper, we determine upper and lower bounds on the cardinality of a maximal non-commuting set in a finite p-group with derived subgroup of prime order.

Finite p-Groups G with H'=G'for Each A2-Subgroup H^(*)

A finite p-group G is called an At-group if t is the minimal non-negative integer such that all subgroups of index pt of G are abelian.The finite p-groups G with H'=G'for all A2-subgroups H of G are classified completely in this paper.As an application,a problem proposed by Berkovich is solved.

无限正则p-群

对一类无限正则p-进行了研究,得到了一个正则的局部幂零P-群G如果满足|G:(?)_1(G)|<∞,那么G是幂零的且G是可除阿贝尔P-群被有限群的扩张.进而,还研究了一类无限的非正则p-群,但它的所有真商群或者真的无限子群是正则群.在假设这类群存在拟循环子群的情况下,在定理1.2和1.3给出了这类群的结构的刻画.

Schmidt定理的推广

设G是个超Abel（hyperabelian）挠群，若G的亏数≤2的次正规Abel子群都满足子群的极小条件，则C是个可解的Cernikov群，这项工作推广了Schmidt的一个有名结果．

关于非循环群的非幂子群数的下确界(英文)

研究了幂子群的一些性质并且得到非循环群的非幂子群个数的下界为3.

Finite p-Groups all of Whose Subgroups of Index p^(3) are Abelian

Suppose that G is a finite p-group.If all subgroups of index p^(t)of G are abelian and at least one subgroup of index p^(t−1)of G is not abelian,then G is called an A_(t)-group.We useA0-group to denote an abelian group.From the definition,we know every finite non-abelian p-group can be regarded as an A_(t)-group for some positive integer t.A_(1)-groups and A_(2)-groups have been classified.Classifying A_(3)-groups is an old problem.In this paper,some general properties about A_(t)-groups are given.A_(3)-groups are completely classified up to isomorphism.Moreover,we determine the Frattini subgroup,the derived subgroup and the center of every A_(3)-group,and give the number of A_(1)-subgroups and the triple(μ_(0),μ_(1),μ_(2))of every A_(3)-group,whereμi denotes the number of A_(i)-subgroups of index p of A_(3)-groups.

A classification of some regular p-groups and its applications

In this paper we classify regular p-groups with type invariants (e, 1, 1, 1) for e ≥ 2 and (1, 1, 1, 1, 1). As a by-product, we give a new approach to the classification of groups of order p5, p ≥ 5 a prime.

Finite p-groups with a minimal non-abelian subgroup of index p(Ⅱ)

We classify completely three-generator finite p-groups G such that Ф(G)≤Z(G)and|G′|≤p2.This paper is a part of the classification of finite p-groups with a minimal non-abelian subgroup of index p,and solve partly a problem proposed by Berkovich.

Finite p-Groups in Which the Number of Subgroups of Possible Order Is Less Than or Equal to p^3

In this paper, groups of order p^n in which the number of subgroups of possible order is less than or equal to p3 ale classified. It turns out that if p 〉 2, n≥ 5, then the classification of groups of order p^n in which the number of subgroups of possible order is less than or equal to p3 and the classification of groups of order p^n with a cyclic subgroup of index p2 are the same.

Finite p-Groups Whose Abelian Subgroups Have a Trivial Intersection

A subgroup H of a finite group G is called a TI-subgroup if H ∩ H^x = 1 or H for all x ∈ G. In this paper, a complete classification for finite p-groups, in which all abelian subgroups are TI-subgroups, is given.

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.