Derived Lengths of Solvable Groups Having Five Irreducible Character Degrees II

Let G be a solvable group with five character degrees. Suppose that there is some prime p so that G/O ^p (G) is not Abelian. Also, assume that cd(G) contains a degree that is not divisible by p. Under these hypotheses, we show that the derived length of G is at most 4.     

The Number of Irreducible Character Degrees of Solvable Groups Satisfying the One-Prime Hypothesis

Let G be a finite group, and write cd(G) for the set of degrees of irreducible characters of G. We say G satisfies the one-prime hypothesis if whenever a and b are distinct degrees in cd(G), then the greatest common divisor of a and b is either 1 or a prime. We show that if G is a solvable group satisfying the one-prime hypothesis, then |cd(G)|≤9. We also construct a solvable group G satisfying the one-prime hypothesis with |cd(G)|=9 which shows that the bound found in this paper is the best possible bound. Presented by D. Passman Mathematics Subject Classification (2000) 20C15.     

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.     

Groups with Four Character Degrees and Derived Length Four

In this article, we consider solvable groups that have four character degrees and derived length four.     

Solvable Groups Satisfying the Two-prime Hypothesis,Part I

In this paper, finite solvable groups satisfying the “-prime hypothesis” are considered. Specifically, a bound on the number of irreducible character degrees of such a group is obtained when . The general situation is also considered, and generalizations of the -prime hypothesis are analyzed. Presented by A. Verschoren.     

特征标次数图是一种连通图的单群

对有限单群G,假设其不可约特征标次数图Δ(G)连通,且图顶点集ρ(G)=π_1∪π_2∪{p},其中|π_1|,|π_2|≥1,π_1∩π_2=θ,且π_1与π_2中顶点不相邻.证明了Δ(G)满足上面的假设的有限单群G只有4种:M_(11),J_1,PSL_3(4)或²B_2(q2B_2(q²),其中q2),其中q²一1是Mersenne素数.     

可解群的$\cd(G)-1$个特征标次数构成的图

Let G be a group. We consider the set cd（G）／{m}, where m ∈ cd（G）. We define the graph △（G - m） whose vertex set is p（G - m）, the set of primes dividing degrees in cd（G）／{m}. There is an edge between p and q in p（G - m） ifpq divides a degree a ∈ cd（G）／{m}. We show that if G is solvable, then △（G - m） has at most two connected components.     

关于具有限秩的可解群

关于具有限秩的可解群本文得到了它的正规列的交换商因子的一种排序,推出了这类群的所有拟循环了群构成它的一个特征子群,了秩ｎ的可解群的Ｈｉｒｓｃｈ不变量≤ｎ,并由此界定了秩ｎ的无挠可解群的导出长度。     

Lie Derived Lengths of Group Algebras of Groups with Cyclic Derived Subgroup

In this article the Lie derived length and the strong Lie derived length of group algebras are determined in the case when the derived subgroup of the basic group is cyclic of odd order. As a consequence, we have the characterization of the group algebras of minimal strong Lie derived length.     

Positions of characters in finite groups and the Taketa inequality

We define the position of an irreducible complex character of a finite group as an alternative to the degree. We then use this to define three classes of groups: position reducible (PR)-groups, inductively position reducible (IPR)-groups and weak IPR-groups. We show that IPR-groups and weak IPR-groups are solvable and satisfy the Taketa inequality (ie, that the derived length of the group is at most the number of degrees of irreducible complex characters of the group), and we show that any M-group is a weak IPR-group. We also show that even though PR-groups need not be solvable, they cannot be perfect.     

Nonsolvable Groups All of Whose Character Degrees are Odd-Square-Free

A finite group G is odd-square-free if no irreducible complex character of G has degree divisible by the square of an odd prime. We determine all odd-square-free groups G satisfying S ≤ G ≤ Aut(S) for a finite simple group S. More generally, we show that if G is any nonsolvable odd-square-free group, then G has at most two nonabelian chief factors and these must be simple odd-square-free groups. If the alternating group A ₇ is involved in G, the structure of G can be further restricted.     

Structure of Solvable Rational Groups

R. Gow proved that the order of a solvable rational group isdivisible only by the primes 2, 3 and 5. In this paper it isproved that in a solvable rational group the Sylow 5-subgroupis always normal and elementary Abelian. Moreover, the structureof rational {2, 5}-groups is described in detail. 2000 MathematicsSubject Classification 20C15, 20C20, 20E34, 20E45.     

On Lengths of Conjugacy Classes and Character Degrees in Finite Groups

We extend the two results of Dolfi and Isaacs to π-separable groups, where π = {p, q}, from solvable groups.     

特征标次数的重数与可解群结构

非线性不可约特征标次数的重数全部为1的有限群的分类是熟知的.对可解群,本文讨论更一般的,即非线性不可约特征标次数的重数都与群阶互素的有限群的纯群论性质.特别地,得到了非线性不可约特征标次数的重数均小于2p的奇阶群G的分类结果.这里p为群阶|G|的最小素因子.     

On the Character Degree Graph of Solvable Groups, I Three Primes

For any finite solvable group G we show that if three primes dividing the degrees of certain irreducible characters of G are given, then there exists an irreducible character of G with degree divisible by at least two of the given primes. This revised version was published online in June 2006 with corrections to the Cover Date.     

Bounding the Derived Length of a Solvable Group: An Improvement on a Result by Gluck

Let cd(G) be the set of degrees of irreducible complex characters and dl(G) be the derived length of the finite group G. It is a result by Gluck that if G is solvable then dl(G) ≤2|cd(G)|. Using a result of Isaacs and Knutson, I will in this article show that this bound can be improved to dl(G) ≤2|cd(G)| ?3 when |cd(G)| ≥3.     

关于可解群p—块的几个结果

主要结果是如下定理:设G是有限可解群使得G/F(G)是奇阶A-群,又设p是一个素数且G不含截断q~(pn):(Z_m:Z_p)。其中q~(pn):(Z_m:Z_n))是初等交换q-群q~(pn)被Z_m:Z_p的扩张,而m=(q~(pn)-1)/(q~n-1)。则G有亏数零p-块的充要条件是O_p(G)=1。     

Orbits of the Actions of Odd-Order Solvable Groups

Suppose that V is a finite faithful irreducible G-module where G is a finite solvable group of odd order. We prove if the action is quasi-primitive, then either F(G) is abelian or G has at least 212 regular orbits on V. As an application, we prove that when V is a finite faithful completely reducible G-module for a solvable group G of odd order, then there exists v ∈ V such that C _G (v) ? F ₂(G) (where F ₂(G) is the 2nd ascending Fitting subgroup of G). We also generalize a result of Espuelas and Navarro. Let G be a group of odd order and let H be a Hall π-subgroup of G. Let V be a faithful G-module over a finite field of characteristic 2, then there exists v ∈ V such that C _H (v) ? O _π(G).     

具有两个p'维非线性不可约特征标的非可解群

McKay猜想是有限群表示理论中的一个重要问题.本文考虑了具有两个p'维不可约特征标的非可解群群G,并证明了McKay猜想对此类群成立.更进一步,当p为奇素数时,下面结果成立:p=3,G≌PSL_2(7)或存在一个正规2-群N,满足G/N≌PSL_2(5).