The Automorphism Group of a Finite p-Group with a Cyclic Frattini Subgroup

Let $G$ be a finite $p$-group with a cyclic Frattini subgroup. Inthis paper, the automorphism group of $G$ is determined.     

一类不能作为自同构群的有限群

本文证明了阶小于ｐ９的有限交换ｐ－群不能作为有限群的自同构群，从而推进了ＭａｃＨａｌｅ的结果．     

On the Relative Commutativity Degree of a Subgroup of a Finite Group

The aim of this article is to give a generalization of the concept of commutativity degree of a finite group G (denoted by d(G)), to the concept of relative commutativity degree of a subgroup H of a group G (denoted by d(H, G)). We shall state some results concerning the new concept which are mostly new or improvements of known results given in Gustafson (1973 Gustafson , W. H. ( 1973 ). What is the probability that two group elements commute? Amer. Math. Monthly 80 : 1031 – 1304 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) and Moghaddam et al. (2005 Moghaddam , M. R. R. , Chiti , K. , Salemkar , A. R. ( 2005 ). n-Isoclinism classes and n-nilpotency degree of finite groups . Algebra Colloquium 12 ( 2 ): 225 – 261 . [Google Scholar]). Moreover, we shall define the relative nth nilpotency degree of a subgroup of a group and give some results concerning this at the end of the article.     

含p^n—2阶元的P^n阶p—群的自同构群

一个有限p-群G称为一个LA-群,如果当G是非循环的且|G|>p~2时有|G|/|Aut(G)|,本文证明了一个含有p~(n-2)阶元的p~n阶p-群是一LA-群。     

On Subgroups Generated by Small Classes in Finite Groups

Let G be a finite group and M(G) be the subgroup of G generated by all noncentral elements of G that lie in the conjugacy classes of the smallest size. Recently several results have been proved regarding the nilpotency class of M(G) and F(M(G)), where F(M(G)) denotes the Fitting subgroup of M(G). We prove some conditional results regarding the nilpotency class of M(G).     

Bounds for the Number of Conjugacy Classes of Non-Normal Subgroups in a Finite p-Group

Let ν(G) be the number of conjugacy classes of non-normal subgroups of a finite group G. We obtain two new lower bounds for ν(G) when G is a non-abelian finite p-group and p is odd. More precisely, if |G| =p ⁿ , exp Z(G) = p ^e , and exp G/G′ =p ^f , let us define λ(G) = n ? e and κ(G) = n ? f. Then we prove that ν(G) ≥ p(λ(G) ?3) +2 and ν(G) ≥ p(κ(G) ?3) +2. The first bound improves the bound ν(G) ≥ λ(G) ?1 given by [10 La Haye , R. , Rhemtulla , A. ( 1999 ). Groups with a bounded number of conjugacy classes of non-normal subgroups . J. Algebra 214 : 41 – 63 .[Crossref], [Web of Science ®] , [Google Scholar]], and almost in every case, the second one improves the bound ν(G) ≥ p(k ? 1) +1 obtained by [6 Fernández-Alcober , G. A. , Legarreta , L. ( 2008 ). Conjugacy classes of non-normal subgroups in finite nilpotent groups . J. Group Theory 11 ( 3 ): 381 – 397 .[Crossref] , [Google Scholar]], where k is defined by the condition that |G′| =p ^k .     

Automorphism Groups of Finite Abelian p-groups and Transvections

We define a particular type of automorphisms called transvections on a finite finite abelian p-group H_p. It is proved that the subgroup E of the automorphism group Aut(H_p) of H_p generated by those transvections is normal in it, and that Aut(H_p) can be written as the product of E and some abelian subgroup K. The center of Aut(H_p) is also determined.     

On Finite p-Groups Whose Central Automorphisms are all Class Preserving

We obtain certain results on a finite p-group whose central automorphisms are all class preserving. In particular, we prove that if G is a finite p-group whose central automorphisms are all class preserving, then d(G) is even, where d(G) denotes the number of elements in any minimal generating set for G. As an application of these results, we obtain some results regarding finite p-groups whose automorphisms are all class preserving.     

子群的核平凡或正规闭包极大的有限p群

称有限p群G为ACT群,如果对每个交换子群H,其正规核HG=1或HG=H.又称p群G是CC群,如果对每个非正规交换子群H,有HG=1或HG在G中的指数为p.本文分类了ACT群和CC群.     

The modular group algebras of p-groups of maximal class II

The modular isomorphism problem is settled for 3-groups of maximal class but two families of groups. Moreover, the conjecture that the ideals belonging to the lower central series of a group base are determined by the structure of the group algebra is refuted in greatest generality by virtue of a single group of order 81 of maximal class, and it is proved that the nilpotency class is determined by the structure of the group algebra for p-groups of maximal class.     

On the Existence of 3-Generator, 3-Relation Finite 2-Groups with a Given (Co)Class

In this article, we show that for any positive integer k there is a 3-generator, 3-relation finite 2-group of class (respectively, coclass) k provided that k ≥ 4 (respectively, k ≥ 5).     

Some 3-Generator, 3-Relation Finite 2-Groups

Using computational methods, we first show that there are exactly eighteen 3-generator 2-groups of order 2¹⁰ with trivial Schur multiplier all having deficiency zero. We next generalize one of the groups obtained to exhibit two infinite classes of 3-generator, 3-relation finite 2-groups of high nilpotency class providing an affirmative answer to a problem posed by Havas et al.     

中心循环的有限p-群中的非交换集

设G是一个群,X是G的一个子集,若对于任意x,y∈X且x≠y,都有xy≠yx,则称X是G的一个非交换集.进一步,如果对于G中的任意其他非交换子集Y,都有|X|≥|Y|,那么称X是G的一个极大非交换集.本文界定了中心循环的有限p-群中极大非交换集的势.     

On Central Automorphisms Fixing the Center Element-Wise

Let G be a finite p-group of nilpotency class 2. We find necessary and sufficient conditions on G such that each central automorphism of G fixes the center of G element-wise.     

On a Generalization of Hamiltonian Groups and a Dualization of PN-Groups

Baer and Wielandt in 1934 and 1958, respectively, considered that the intersection of the normalizers of all subgroups of G and the intersection of the normalizers of all subnormal subgroups of G. In this article, for a finite group G, we define the subgroup S(G) to be intersection of the normalizers of all non-cyclic subgroups of G. Groups whose noncyclic subgroups are normal are studied in this article, as well as groups in which all noncyclic subgroups are normalized by all minimal subgroups. In particular, we extend the results of Passman, Bozikov, and Janko to non-nilpotent finite groups.     

Finite Groups Having Exactly Two Conjugate Classes of Non-subnormal Subgroups

In this paper, finite groups having exactly two conjugate classes of non-subnormal subgroups of G are completely classified.     

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 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.     

On the Wielandt Subgroup in a p-Group of Maximal Class

The Wielandt subgroup of a group G,denoted by w(G),is the intersection of the normalizers of all subnormal subgroups of G.In this paper,the authors show that for a p-group of maximal class G,either wi(G) = ζi(G) for all integer i or wi(G) = ζi+1(G) for every integer i,and w(G/K) = ζ(G/K) for every normal subgroup K in G with K = 1.Meanwhile,a necessary and suflcient condition for a regular p-group of maximal class satisfying w(G) = ζ2(G) is given.Finally,the authors prove that the power automorphism group PAut(G) is an elementary abelian p-group if G is a non-abelian pgroup with elementary ζ(G) ∩ 1(G).