On the Finite Group with the Index of Maximal Subgroups of the Monster

Abstract

We study the problem concerning the influence of the index of maximal subgroup or the degree of primitive permutation representation of the finite simple groups on the structure of a group. Let G be a finite group and s be the index of maximal subgroup of the Monster M. In this paper, we prove that there exists an epimorphism from G to M or A _s if G has the primitive permutation representation of degree s, and as a consequence we prove that the Monster is determined by every s.     

与小魔群B有共同本原置换表示次数的有限群

本文研究小魔群B的极大子群的指数对群的结构的影响。设G是有限群,π_t(G)和π_t(B)分别表示G和B的极大子群的指数集,s_(26)=[B:M_(11)]。设π_t(B)∩π_t(G)≠φ,对任意s ∈ π_t(B)∩π_t(G),如果s≠s_(26),那么G与B或A_s同态;如果s=s_(26),那么G与B或A_1(s_(26)—1)或A_s同态。     

闭2-极大子群指数为素数幂的有限群

设G为有限群, 如果G的每个非2-闭极大子群的指数均为素数幂, 那么G-S(G)≌PSL2(7)或1, 其中S(G)为G的最大可解正规子群.     

A characterization ofL n (q) by the normalizers' orders of their sylow subgroups

LetG be a finite group. If for every primer, whereR ₁ Syl _r G andR ₂ Syl _r (L _n (q)), thenG L _n (q).     

有限群的s-条件置换子群

如果对群G的任意Sylow子群T,存在一个元素x∈G,使得HT~x=T~xH,那么称群G的子群H在G中s-条件置换．利用s-条件置换子群给出了一些群的性质和结构．     

A characterization of the finite simple groups with set of indices of their maximal subgroups

We study the problem concerning the influence of indices of maximal subgroups of a simple group on the structure of a group. We obtain a characterization property of all finite simple groups.     

Maximal Subgroups of Finite Orthogonal Groups Stabilizing Spreads of Lines

We prove that, for q odd and n ≥ 3, the group G = O _n (q ²) · 2 is maximal in either the orthogonal group O _2n (q) or the special orthogonal group SO _2n (q). The group G corresponds to the stabilizer of a spread of lines of PG(2n ? 1, q) in which some lines lie on a quadric, some are secant to the quadric, and others are external to the quadric.     

On locally primitive Cayley graphs of finite simple groups

In this paper we investigate locally primitive Cayley graphs of finite nonabelian simple groups. First, we prove that, for any valency d for which the Weiss conjecture holds (for example, d?20 or d is a prime number by Conder, Li and Praeger (2000) [1]), there exists a finite list of groups such that if G is a finite nonabelian simple group not in this list, then every locally primitive Cayley graph of valency d on G is normal. Next we construct an infinite family of p-valent non-normal locally primitive Cayley graph of the alternating group for all prime p?5. Finally, we consider locally primitive Cayley graphs of finite simple groups with valency 5 and determine all possible candidates of finite nonabelian simple groups G such that the Cayley graph Cay(G,S) might be non-normal.     

有限群的弱S-拟正规嵌入子群

群G的子群H称为在G中S-拟正规嵌入的,如果对于任意的素数p||H|,H的Sylow p-子群也是G的某个S-拟正规子群的Sylow p-子群.称群G的子群H在G中弱S-拟正规嵌入,如果存在群G的正规子群T,使得HT■G且H∩T在G中是S-拟正规嵌入的.研究了弱S-拟正规嵌入子群的性质,给出了某些群类的新的特征,并推广了一些已知的结论.     

On maximal subgroups of the multiplicative group of a division algebra

The question of existence of a maximal subgroup in the multiplicative group D^* of a division algebra D finite-dimensional over its center F is investigated. We prove that if D^* has no maximal subgroup, then deg(D) is not a power of 2, F^*2 is divisible, and for each odd prime p dividing deg(D), there exist noncyclic division algebras of degree p over F.     

Finite groups with S-permutable n-minimal subgroups

We study the supersolvability of finite groups and the nilpotent length of finite solvable groups under the assumption that all their exactly n-minimal subgroups are S-permutable, where n is an arbitrary integer.     

Carter subgroups of finite almost simple groups

In the paper we work to complete the classification of Carter subgroups in finite almost simple groups. In particular, it is proved that Carter subgroups of every finite almost simple group are conjugate. Based on our previous results, together with those obtained by F. Dalla Volta, A. Lucchini, and M. C. Tamburini, as a consequence we derive that Carter subgroups of every finite group are conjugate. Supported by RFBR grant No. 05-01-00797; by the Council for Grants (under RF President) for Support of Young Russian Scientists via projects MK-1455.2005.1 and MK-3036.2007.1; by SB RAS Young Researchers Support grant No. 29; via Integration Project No. 2006.1.2. __________ Translated from Algebra i Logika, Vol. 46, No. 2, pp. 157–216, March–April, 2007.