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1.
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.  相似文献   

2.
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.  相似文献   

3.
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)={1,2,2e+1,2e+1,2(e+1)},并给出了d>1时群的结构.  相似文献   

4.
5.
Ni Du 《代数通讯》2013,41(11):4660-4673
In this article, we consider solvable groups that have four character degrees and derived length four.  相似文献   

6.
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.  相似文献   

7.
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.  相似文献   

8.
关于具有限秩的可解群   总被引:4,自引:0,他引:4  
刘合国 《数学进展》2000,29(1):55-60
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。  相似文献   

9.
Zsolt Balogh 《代数通讯》2013,41(2):315-324
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.  相似文献   

10.
    
Tobias Kildetoft 《代数通讯》2017,45(6):2325-2333
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.  相似文献   

11.
Mark L. Lewis 《代数通讯》2013,41(4):1273-1292
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.  相似文献   

12.
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.  相似文献   

13.
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.  相似文献   

14.
张继平 《数学进展》1993,22(2):133-138
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。  相似文献   

15.
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.  相似文献   

16.
In this paper, we study two classes of 2-generated 2-groups of nilpotency class 2 classified by Kluempen in 2002 and also a class of finite 2-groups of high nilpotency class for their Fibonacci lengths. Their involvement in certain interesting sequences of Tribonacci numbers gives us some explicit formulas for the Fibonacci lengths and this adds to the small class of finite groups for which the Fibonacci length are known.  相似文献   

17.
We study the Berezin transform in the context of solvable groups AN (acting on homogeneous cones and Siegel domains) and determine its spectral decomposition, using an explicit integral kernel representation for the associated eigen-operators in terms of multivariable hypergeometric functions.  相似文献   

18.
The structure of finite solvable groups in which any Sylow subgroup is the product of two cyclic subgroups is studied. In particular, it is proved that the nilpotent length of such a group is no greater than 4. It is also proved that the nilpotent length of a finite solvable group in which the index of any maximal subgroup is either a prime or the square of a prime or the cube of a prime does not exceed 5.  相似文献   

19.
《代数通讯》2013,41(9):3391-3402
Abstract

Let G be a finite, nonabelian, solvable group. Following work by D. Benjamin, we conjecture that some prime must divide at least a third of the irreducible character degrees of G. Benjamin was able to show the conjecture is true if all primes divide at most 3 degrees. We extend this result by showing if primes divide at most 4 degrees, then G has at most 12 degrees. We also present an example showing our result is best possible.  相似文献   

20.
Jinbao Li 《代数通讯》2013,41(7):2971-2983
In the past thirty years, several kinds of quantitative characterizations of finite groups especially finite simple groups have been investigated by many mathematicians. Such as quantitative characterizations by group order and element orders, by element orders alone, by the set of sizes of conjugacy classes, by dimensions of irreducible characters, by the set of orders of maximal abelian subgroups and so on. Here the authors continue this topic in a new area tending to characterize finite simple groups with given orders by some special conjugacy class sizes, such as largest conjugacy class sizes, smallest conjugacy class sizes greater than 1 and so on.  相似文献   

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