关于群的阶与共轭类数

讨论群的共轭数与群阶的关系,获得两个新的数量不等式,同时改进了一些相关的已知结果。     

恰有两个子群共轭类长的有限群

设G是有限群,Ns(G)表示G的子群共轭类长构成的集合.本文研究Ns(G)中只有两个元素时有限群G的结构,在非幂零情形时给出了G的完全分类,在幂零情形时获得了G的一些性质.     

论有限群元素共轭类的平均长度

本文研究有限群元素共轭类的平均长度问题.利用初等群论方法和有限群特征标理论,在共轭类平均长度为某一定数时,获得了对有限群结构的刻划,且对有限群数量性质的研究是有意义的.     

Finite Groups with Four Conjugacy Class Sizes

We determine the structure of all finite groups with four class sizes when two of them are coprime numbers larger than 1. We prove that such groups are solvable and that the set of class sizes is exactly {1, m, n, mk}, where m, n > 1 are coprime numbers and k > 1 is a divisor of n.     

On a Commuting Graph on Conjugacy Classes of Groups

We consider the graph Γ(G), associated with the conjugacy classes of a group G. Its vertices are the nontrivial conjugacy classes of G, and we join two different classes C, D, whenever there exist x ∈ G and y ∈ D such that xy = yx. The aim of this article is twofold. First, we investigate which graphs can occur in various contexts and second, given a graph Γ(G) associated with G, we investigate the possible structure of G. We proved that if G is a periodic solvable group, then Γ(G) has at most two components, each of diameter at most 9. If G is any locally finite group, then Γ(G) has at most 6 components, each of diameter at most 19. Finally, we investigated periodic groups G with Γ(G) satisfying one of the following properties: (i) no edges exist between noncentral conjugacy classes, and (ii) no edges exist between infinite conjugacy classes. In particular, we showed that the only nonabelian groups satisfying (i) are the three finite groups of order 6 and 8.     

Finite Groups with a Disconnected p -Regular Conjugacy Class Graph

Abstract

Let G be a finite p-solvable group for a fixed prime p. Attach to G a graph Γ _p (G) whose vertices are the non-central p-regular conjugacy classes of G and connect two vertices by an edge if their cardinalities have a common prime divisor. In this note we study the structure and arithmetical properties of the p-regular class sizes in p-solvable groups G having Γ _p (G) disconnected.     

素数幂、双素数幂阶元的共轭类长的个数为4的 有限群的结构

设群G为一个有限群.如果群G中素数幂、双素幂阶元的共轭类长的集合为{1,p~a,m,p~bm},那么群G是可解的,其中ab为正整数,p为素数且与m互素.进一步,给出了群G/Z(G)的结构,这是对文"Chen R F,Zhao X H.A criterion for a group to have nilpotent p-complements[J].Monatsh Math,2016,179(2):221-225"中定理A主要结论的一个推广.     

幂零群中非正规循环子群的共轭类数

设G是有限幂零群,v~*(G)是其非正规循环子群的共轭类数,则下列结论之一成立：(1) v~*(G)≥c(G)-1；(2)G是Hamilton群；(3)G中存在正规子群K使K/Z(K)有一个同态像与二面体群D(2~n),n≥3或C_2×C_2同构．     

On the Diameter of a p-Regular Conjugacy Class Graph of Finite Groups II

Abstract

Let Gbe a finite p-solvable group. Let us consider the graph Γ* _p (G) whose vertices are the primes which occur as the divisors of the conjugacy classes of p-regular elements of G and two primes are joined by an edge if there exists such a class whose size is divisible by both primes. Suppose that Γ _p *(G) is a connected graph, then we prove that the diameter of this graph is at most 3 and this is the best bound.     

共轭类的长和有限群的结构

设Ｇ是有限群，并设Ｃｏｎ（Ｇ）是由Ｇ的一切共轭类组成的集合。本文目的是考察Ｃｏｎ（Ｇ）的算术结构对Ｇ的群结构的影响。我们着重于Ｃｏｎ（Ｇ）的ｐ－部分结构，并得到关于ｐ－幂零群的两个定理。这里，ｐ表示一个有限群的最小素因子。用这两个定理，我们还得到若干结果，其中两个改进了Ｄ．Ｃｈｉｌｌａｇ和Ｍ．Ｈｅｒｚｏｇ关于共轭类长的两个结果。     

3阶Feigenbaum映射的拓扑共轭性

本文讨论3阶Feigenbaum映射限制在非游荡集上的拓扑共轭性．一方面3阶Feigenbaum映射必然产生混沌,混沌的产生使得非游荡集复杂化;另一方面3阶Feigenbaum映射又分为单谷的和非单谷的两类．利用有限型子转移,证明了对任意给定的两个满足一定条件的3阶Feigenbaum映射,限制在其非游荡集上是拓扑共轭．     

有限群的共轭类个数与群的性质

用群的共轭类个数刻画了交换群,同时用一个很简洁的方法重新证明了Frobenius G提出的一个著名问题:对于一个固定的数自然数n,共轭类数为n的有限群,在同构的意义下是有限的.     

Coxter群中二阶元的判定及其在仿射Weyl群中的应用

本文首先给出Coxter群中二阶元的判定方法，应用到E6型有限wey1群及A21^(1)型仿射Wey1群，给出了其二阶元的共轭标准形与共轭类数．     

Conjugacy problem in a class of 2-groups

In the paper, the conjugacy problem for elements of Grigorchuk 2-groups is solved. Each group is regarded from the viewpoint of rearrangement-like transformations of the interval (0, 1) and from the viewpoint of wreath products of groups. Several approaches are indicated that allow one to establish conjugacy conditions of elements.Translated fromMatematicheskie Zametki, Vol. 64, No. 4, pp. 573–583, October, 1998.The author wishes to express his deep gratitude to V. I. Sushchanskii and M. N. Vasyakina for their support and attention to the research.     

有限群共轭类长的一个问题

设A和B都是有限群G的子群且G=AB.若A是G的次正规子群,且对每个p∈π(G)以及每个素数幂阶的p′-元x∈A∪B,p~2均不整除|x~G|,则G为超可解群.这个结果正面解答了由石向东,韦华全和马儇龙于2013年提出的一个问题,统一推广了由刘晓蕾于2011年得到的三个定理.     

Conjugacy Classes in Sylow p-Subgroups of Finite Chevalley Groups in Bad Characteristic

Let U = U(q) be a Sylow p-subgroup of a finite Chevalley group G = G(q). Röhrle and Goodwin in 2009 determined a parameterization of the conjugacy classes of U, for G of small rank when q is a power of a good prime for G. As a consequence they verified that the number k(U) of conjugacy classes of U is given by a polynomial in q with integer coefficients. In the present paper, we consider the case when p is a bad prime for G. Our motivation is to observe how the situation differs between good and bad characteristics. We obtain a parameterization of the conjugacy classes of U, when G has rank less than or equal to 4, and G is not of type F ₄. In these cases we deduce that k(U) is given by a polynomial in q with integer coefficients; this polynomial is different from the polynomial for good primes.     

The Prime Graph of a Sporadic Simple Group

Abstract

Let Gbe a finite group and Sa sporadic simple group. We denote by π(G) the set of all primes dividing the order of G. The prime graph Γ(G) of Gis defined in the usual way connecting pand qin π(G) when there is an element of order pqin G. The main purpose of this paper is to determine finite group Gsatisfying Γ(G) = Γ(S) (See Theorem 3) and to give applications which generalize Abe (Abe, S. Two ways to characterize 26 sporadic finite simple groups. Preprint) and Chen (Chen, G. (1996). A new characterization of sporadic simple groups. Algebra Colloq.3:49–58). The results are elementary but quite useful.