特征标次数的商与有限群结构

本文考察特征标次数的商如何影响有限群的结构.设G是非线性不可约特征标次数的商都是n次方自由的有限群,首先,对可解群G证明了它的导长能被一个仅依赖于n的函数所界定;其次,给出了当n≤3时有限群G的结构.     

特征标次数的商与有限群结构

本文考察特征标次数的商如何影响有限群的结构.设G是非线性不可约特征标次数的商都是n次方自由的有限群,首先,对可解群G证明了它的导长能被一个仅依赖于n的函数所界定;其次,给出了当n≤3时有限群G的结构.     

特征标次数素图中不含三角形的单群

讨论了不可约特征标次数素图中不含三角形的单群.证明了:若G是有限单群且其素图中不含三角形,则G(≌)L2(q),其中q≥4,且满足条件|π(q+1)| ≤ 2和|π(q-1)|≤2.     

有限群特征标次数商的几点注记

对自然数n ,W(n)表示n中的素因子个数 (计重数 ) .对于有限群G的不可约复特征标集Irr(G) ,令W0 (G) =max{W(|G :kerχ|χ(1) ) | χ∈Irr(G) ,χ(1) >1} ,本文将考察W0 (G)≤ 3时有限群G的群论结构 .     

有限群的特征标余次数

对于有限群G的一个不可约特征标χ,我们定义它的余次数为cod(χ)=|G:kerχ|/χ(1).本文介绍有限群特征标余次数方面的工作进展和未解决的问题.     

关于有限群的不可约特征标个数的一个结果

本文研究了一般的非交换有限群G的阶与不可约特征标个数的比值与群G结构之间的关系.通过群G阶的最小素因子和G的换位子群的阶的最小素因子,得出了这个比值的下界,并确定了达到下界的一个充分必要条件.     

关于Brauer特征标扩张的一点注记

研究有限群特征标可扩张的情况是有限群表示论领域中一个有意义的问题.设G为有限群,用Irr（G）表示G的所有不可约复特征标构成的集合.设N（？）G,θ∈Irr（N）且θ是G-不变.如果（｜G/N｜,o（θ）θ（1））=1,则[1]中的推论8.16说明了此时υ到G有唯一的扩张χ,且o（χ）=o（θ）.此结论启发了我们可以从特征标的行列式阶的角度来思考特征标扩张的情形.本文将利用有限群Brauer特征标的行列式阶,着重考虑Brauer特征标的可扩张情形.另外我们也得到了一个关于Brauer特征标次数的结论.     

具有TI亏群块的Alperin猜想问题

设P是一个素数，G是一个有限群B是G的一个p－块，其亏群为TI子群B是B在Brauer第一主要定理下的对应块．本文证明如下等价条件：（1）B和B有相同的常不可约特征标数；（2）B和B有相同的模不可约特征标数；（3）B和B的Cartan矩阵有相同重数的1作为它们的不变因子数；（4）Alperin猜想对B成立．     

非SM-群的一类rp~3-阶群

称有限群的不可约特征标x为SM-特征标,如果x可由某个次正规子群的线性特征标诱导得到.称有限群为SM-群,如果有限群的所有不可约特征标均为SM-特征标.通过一个例子,将说明rp~3-阶群不一定是SM-群.     

非线性不可约特征标维数相等且为素数个的有限群

对于具有素数个非线性不可约特征标且它们的维数相等的有限群,我们给出一个分类.     

有限群的特征标的零点和可解φ-群

有两个对偶的问题如下：问题Ⅰ：将满足下述条件的有限群G分类：G的特征标表中，除一行外其余各行最多有一个零．问题Ⅱ：将满足下述条件的有限群G分类：G的特征标表中，除一列外其余各列最多有一个零．在这篇文章中，我们对于有限可解群解答上述两个问题，并确定和这两个问题密切相关的一类有限可解群的结构（这类可解群在本文中称之为可解φ-群）．附带我们还完全回答了[4]中的问题1，并说明[6，定理]的条件可以极大地减弱．     

Orders generated by character values

Let $$K:=\mathbb {Q}(G)$$ be the number field generated by the complex character values of a finite group G. Let $$\mathbb {Z}_K$$ be the ring of integers of K. In this paper we investigate the suborder $$\mathbb {Z}[G]$$ of $$\mathbb {Z}_K$$ generated by the character values of G. We prove that every prime divisor of the order of the finite abelian group $$\mathbb {Z}_K/\mathbb {Z}[G]$$ divides |G|. Moreover, if G is nilpotent, we show that the exponent of $$\mathbb {Z}_K/\mathbb {Z}[G]$$ is a proper divisor of |G| unless $$G=1$$. We conjecture that this holds for arbitrary finite groups G.     

子群为拟正规或自正规的有限群

本文研究了每个子群为拟正规或自正规的有限群，给出了这类群的完全分类，主要结果为定理Ｇ的每个子群为拟正规或自正规当且仅当Ｇ为下列情形之一：Ⅰ）Ｇ为拟Ｈａｍｉｌｔｏｎ群，Ⅱ）Ｇ＝ＨＰ，其中Ｈ为Ｇ的正规ａｂｅｌｉａｎｐ＇－Ｈａｌｌ子群．Ｐ＝〈ｘ〉∈Ｓｙｌ＿ｐ（Ｇ）。〈ｘ￣ｐ〉＝Ｏ＿ｐ（Ｇ）＝Ｚ（Ｇ），ｘ在Ｈ上诱导Ｈ的一个ｐ阶无不动点的幂自同构．ｐ为｜Ｇ｜的最小素因子。由此定理可得文［１］所获得的定理。     

Solvable D_2-Groups

Let G be a finite group, Irr_1(G) be the set of nonlinear irreducible characters of G and cd_1(G) the set of degrees of the characters in Irr_1(G). A group G is said to be a D_2-group if |cd_1(G)| =|Irr_1(G)|-2. In this paper, we give a complete classification of solvable D_2-groups.     

Solvable <Emphasis Type="Italic">D</Emphasis><Subscript>2</Subscript>-groups

Let G be a finite group, Irr¹(G) be the set of nonlinear irreducible characters of G and cd¹(G) the set of degrees of the characters in Irr¹(G). A group G is said to be a D₂-group if|cd¹(G)|=|Irr¹(G)|-2. In this paper, we give a complete classification of solvable D₂-groups.     

Nonsolvable D 2-groups

Let G be a finite group. Let Irr₁(G) be the set of nonlinear irreducible characters of G and cd₁(G) the set of degrees of the characters in Irr₁(G). A group G is said to be a D₂-group if |cd₁(G)| = |Irr₁(G)| - 2. The main purpose of this paper is to classify nonsolvable D₂-groups.     

Simple exceptional groups of Lie type are determined by their character degrees

Let G be a finite group. Denote by Irr(G) the set of all irreducible complex characters of G. Let ${{\rm cd}(G)=\{\chi(1)\;|\;\chi\in {\rm Irr}(G)\}}$ be the set of all irreducible complex character degrees of G forgetting multiplicities, and let X₁(G) be the set of all irreducible complex character degrees of G counting multiplicities. Let H be any non-abelian simple exceptional group of Lie type. In this paper, we will show that if S is a non-abelian simple group and ${{\rm cd}(S)\subseteq {\rm cd}(H)}$ then S must be isomorphic to H. As a consequence, we show that if G is a finite group with ${{\rm X}_1(G)\subseteq {\rm X}_1(H)}$ then G is isomorphic to H. In particular, this implies that the simple exceptional groups of Lie type are uniquely determined by the structure of their complex group algebras.     

Simple exceptional groups of Lie type are determined by their character degrees

Let G be a finite group. Denote by Irr(G) the set of all irreducible complex characters of G. Let cd(G)={c(1)  |  c ? Irr(G)}{{\rm cd}(G)=\{\chi(1)\;|\;\chi\in {\rm Irr}(G)\}} be the set of all irreducible complex character degrees of G forgetting multiplicities, and let X₁(G) be the set of all irreducible complex character degrees of G counting multiplicities. Let H be any non-abelian simple exceptional group of Lie type. In this paper, we will show that if S is a non-abelian simple group and cd(S) í cd(H){{\rm cd}(S)\subseteq {\rm cd}(H)} then S must be isomorphic to H. As a consequence, we show that if G is a finite group with X₁(G) í X₁(H){{\rm X}_1(G)\subseteq {\rm X}_1(H)} then G is isomorphic to H. In particular, this implies that the simple exceptional groups of Lie type are uniquely determined by the structure of their complex group algebras.     

不可约特征标次数为等差数的有限可解群

设G为有限群,cd(G)表示G的所有复不可约特征标次数的集合.本文研究了不可约特征标次数为等差数的有限可解群,得到两个结果:如果cd(G)={1,1+d,1+2d,…,1+kd},则k≤2或cd(G)={1,2,3,4};如果cd(G)={1,a,a+d,a+2d,…,a+kd},|cd(G)|≥4,(a,d)=1,则cd(G)={1,2,2^e+1,2e+1,2^(e+1)},并给出了d>1时群的结构.     

Maximal orders of Abelian subgroups in finite simple groups

We bring out upper bounds for the orders of Abelian subgroups in finite simple groups. (For alternating and classical groups, these estimates are, or are nearly, exact.) Precisely, the following result, Theorem A, is proved. Let G be a non-Abelian finite simple group and G L₂ (q), where q=p^t for some prime number p. Suppose A is an Abelian subgroup of G. Then |A|³<|G|. Our proof is based on a classification of finite simple groups. As a consequence we obtain Theorem B, which states that a non-Abelian finite simple group G can be represented as ABA, where A and B are its Abelian subgroups, iff G≌ L₂(2^t) for some t ≥ 2; moreover, |A|-2^t+1, |B|=2^t, and A is cyclic and B an elementary 2-group. Translated fromAlgebra i Logika, Vol. 38, No. 2, pp. 131–160, March–April, 1999.