Groups with π-maximal and π-layer maximal conditions

By analogy with π-minimal, π-layer minimal, primary minimal, and layer minimal conditions, similar maximal conditions are introduced and studied. Translated fromMatematicheskie Zametki, Vol. 68, No. 2, pp. 311–320, August, 2000.     

Characterization of groups with generalized chernikov periodic part

A characterization of groups with generalized Chernikov periodic part is obtained first in the class of groups without elements of order 2 and then without this restriction. Two author’s theorems for periodic groups are generalized to mixed groups. Translated fromMatematicheskie Zametki, Vol. 67, No. 2, pp. 270–275, February, 2000.     

The Average Sylow Multiplicity Character and the Solvable Residual

Let G be a finite group, and let p₁,…, p_m be the distinct prime divisors of |G|. Given a sequence 𝒫 =P₁,…, P_m, of Sylow p_i-subgroups of G, and g ∈ G, denote by m_𝒫(g) the number of factorizations g = g₁…g_m such that g_i ∈ P_i. The properly normalized average of m_𝒫 over all 𝒫 is a complex character over G whose kernel contains the solvable radical of G [7 Levy , D. ( 2010 ). The average Sylow multiplicity character and solvability of finite groups . Communications in Algebra. 38 : 632 – 644 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]]. The present paper characterizes the solvable residual of G in terms of this character.     

Norm商群的Sylow子群皆循环的有限群

设$G$是有限群, $N(G)$为$G$的norm, 则$N(G)$是$G$的正规化G的每个子群的特征子群. 我们在下列条件之一下,研究了$G$的结构：1) Norm商群$G/N(G)$是循环群;2) Norm商群$G/N(G)$的所有Sylow子群都是循环群,特别地当$G/N(G)$的阶是无平方因子数时.     

The L p Spectrum of Locally Symmetric Spaces with Small Fundamental Group

We determine the L ^p spectrum of the Laplace-Beltrami operator on certain complete locally symmetric spaces M whose universal covering X is a symmetric space of non-compact type with rank one. More precisely, we show that the L ^p spectra of M and X coincide if the fundamental group of M is small and if the injectivity radius of M is bounded away from zero. In the L ² case, the restriction on the injectivity radius is not needed.      

关于有限群的$CAP$-嵌入子群

A subgroup H of a finite group G is said to be CAP-embedded subgroup of G if, for each prime p dividing the order of H, there exists a CAP-subgroup K of G such that a Sylow p-subgroup of H is also a Sylow p-subgroup of K. In this paper some new results are obtained based on the assumption that some subgroups of prime power order have the CAP-embedded property in the group.     

The influence of X-semipermutability of subgroups on the structure of finite groups

Let A be a subgroup of a group G and X be a nonempty subset of G. A is said to be X-semipermutable in G if A has a supplement T in G such that A is X-permutable with every subgroup of T. In this paper, we investigate further the influence of X-semipermutability of some subgroups on the structure of finite groups. Some new criteria for a group G to be supersoluble or p-nilpotent are obtained.     

The influence of <Emphasis Type="Italic">X</Emphasis>-semipermutability of subgroups on the structure of finite groups

Let A be a subgroup of a group G and X be a nonempty subset of G. A is said to be X-semipermutable in G if A has a supplement T in G such that A is X-permutable with every subgroup of T. In this paper, we investigate further the influence of X-semipermutability of some subgroups on the structure of finite groups. Some new criteria for a group G to be supersoluble or p-nilpotent are obtained. This work was supported by National Natural Science Foundation of China (Grant Nos. 10771172, 10771180)     

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.