One-Factorizations of Complete Graphs with a Doubly Transitive Automorphism Group

It is shown that a 1-factorization of K_n with a doubly transitiveautomorphism group on vertices is either the affine line-parallelismof AG(d, 2), or one of three ‘sporadic’ exampleswith n = 6, 12 or 28. The full automorphism groups are respectivelyAGL (d, 2) (the holomorph of an elementary abelian group oforder 2^d), PGL(2,5), PSL(2,11) and PL(2,8).     

Doubly Transitive Automorphism Groups of Combinatorial Surfaces

   Abstract. The combinatorial surfaces with doubly transitive automorphism groups are classified. This is established by classifying the automorphism groups of combinatorial surfaces which act doubly transitively on the vertices of the surface. The doubly transitive automorphism groups of combinatorial surfaces are the symmetric group S ₄ , the alternating group A ₅ and the Frobenius group C ₇ · C ₆ . In each case the combinatorial surface is uniquely determined. The symmetric group S ₄ acts doubly transitively on the tetrahedron surface, the alternating group A ₅ on the triangulation of the projective plane with six vertices and the Frobenius group C ₇ · C ₆ on the Moebius torus with seven vertices.     

Doubly Transitive Automorphism Groups of Combinatorial Surfaces

Abstract. The combinatorial surfaces with doubly transitive automorphism groups are classified. This is established by classifying the automorphism groups of combinatorial surfaces which act doubly transitively on the vertices of the surface. The doubly transitive automorphism groups of combinatorial surfaces are the symmetric group S ₄ , the alternating group A ₅ and the Frobenius group C ₇ · C ₆ . In each case the combinatorial surface is uniquely determined. The symmetric group S ₄ acts doubly transitively on the tetrahedron surface, the alternating group A ₅ on the triangulation of the projective plane with six vertices and the Frobenius group C ₇ · C ₆ on the Moebius torus with seven vertices.     

关于全线性群的极大子群和自同构

本文先给出域F上n维线性空间V_n(F)中r维子空间W在全线性群GLn(F)中的稳定子群H=stab(W)的一个群论刻划,然后再利用这一刻划定出GL_n(F)的全部自同构。为叙述方便,下面我们把一个群中的非周期元素的阶规定为零。     

l-群的极大下子群

在非正规值的条件下,给出l-群极大下子群的唯一表达形式,由此推广[1]中的一个结果.与此同时,建立了正规值l-群,特殊值l-群及有限值l-群之间等价的一个充要条件.     

Locally Transitive Graphs Admitting a Group with Cyclic Sylow Subgroups

All graphs are finite simple undirected and of no isolated vertices in this paper. Using the theory of coset graphs and permutation groups, it is completed that a classification of locally transitive graphs admitting a non-Abelian group with cyclic Sylow subgroups. They are either the union of the family of arc-transitive graphs, or the union of the family of bipartite edge-transitive graphs.     

Supersolubility of a Finite Group with Normally Embedded Maximal Subgroups in Sylow Subgroups

Let P be a subgroup of a Sylow subgroup of a finite group G. If P is a Sylow subgroup of some normal subgroup of G then P is called normally embedded in G. We establish tests for a finite group G to be p-supersoluble provided that every maximal subgroup of a Sylow p-subgroup of X is normally embedded in G. We study the cases when X is a normal subgroup of G, X = O_p',p(H), and X = F*(H) where H is a normal subgroup of G.     

The Automorphism Group of a Class of Nilpotent Groups with Infinite Cyclic Derived Subgroups

The automorphism group of a class of nilpotent groups with infinite cyclic derived subgroups is determined. Let G be the direct product of a generalized extraspecial Z-group E and a free abelian group A with rank m, where E ={(1 kα_1 kα_2 ··· kα_nα_(n+1) 0 1 0 ··· 0 α_(n+2)...............000...1 α_(2n+1)000...01|αi∈ Z, i = 1, 2,..., 2 n + 1},where k is a positive integer. Let AutG G be the normal subgroup of Aut G consisting of all elements of Aut G which act trivially on the derived subgroup G of G, and AutG/ζ G,ζ GG be the normal subgroup of Aut G consisting of all central automorphisms of G which also act trivially on the center ζ G of G. Then(i) The extension 1→ Aut_(G') G→ AutG→ Aut(G')→ 1 is split.(ii) Aut_(G') G/Aut_(G/ζ G,ζ G)G≌Sp(2 n, Z) ×(GL(m, Z)■(Z~)m).(iii) Aut_(G/ζ G,ζ GG/Inn G)≌(Z_k)~(2n)⊕(Z)~(2nm).     

Subgroups of the Spinor Group that Contain a Split Maximal Torus. II

In the first paper of the series, we proved the standardness of a subgroup H containing a split maximal torus in the split spinor group Spin(n,R) over a field K of characteristic different from 2 containing at least 7 elements under one of the following additional assumptions: (1) H is reducible, (2) H is imprimitive, (3) H contains a nontrivial root element. In the present paper, we complete the proof of a result announced by the author in 1990 and prove the standardness of all intermediate subgroups, provided that n=2l and . For an algebraically closed K, this follows from a classical result of Borel and Tits, and for a finite K this was proved by Seitz. Similar results for subgroups of the orthogonal groups SO(n,R) were previously obtained by the author not only for fields, but for any commutative semilocal rings R with residue fields large enough. Bibliography: 52 titles.     

Sum of Element Orders of Maximal Subgroups of the Symmetric Group

For a finite group G, let ψ(G) denote the sum of element orders of G. The aim of this article is to show that ψ(H) < ψ(A _n ) for every proper subgroup H of the symmetric group of degree n, which is different from the alternating group A _n .     

Maximal Divisible Subgroups in p-Mixed Modular Abelian Group Rings

The isomorphism structure of the maximal divisible subgroup dVR(G) of the normed unit group VR(G) in a commutative group ring R(G) is completely described only in terms of R and G whenever R is a commutative unital ring of prime characteristic p and G is a p-mixed abelian group. This somewhat extends a result due to Nachev [13 Nachev , N. A. ( 1995 ). Invariants of the Sylow p-subgroup of the unit group of a commutative group ring of characteristic p. Commun. Algebra 23 : 2469 – 2489 . [Google Scholar]] as well as results due to the author [1 Danchev , P. V. ( 2004 ). Maximal divisible subgroups in modular group algebras of p-mixed and p-splitting abelian groups . Rad. Mat. 13 : 23 – 32 . [Google Scholar]] and Kuneva [11 Kuneva , V. ( 2006 ). Maximal divisible subgroups in modular group algebras of p-mixed abelian groups . Compt. rend. Acad. bulg. Sci. 59 : 899 – 902 . [Google Scholar]].     

On Maximal Unipotent Subgroups of a Special Linear Group Over Commutative Ring

Ukrainian Mathematical Journal - It is proved that all maximal unipotent subgroups of a special linear group over commutative ring with identity (such that the factor ring of its modulo primitive...     

有限群的最大子群的性质对群结构的影响

有限群G的一个子群称为在G中是π-拟正规的若它与G的每一个Sylow-子群是交换的.G的一个子群H称为在G中是c-可补的若存在G的子群N使得G=HN且H∩N≤HG=CoreG(H).本文证明了:设F是一个包含超可解群系u的饱和群系,G有一个正规子群H使得G/H∈F.则G∈F若下列之一成立:(1)H的每个Sylow子群的所有极大子群在G中或者是π-拟正规的或者是c-可补的;(2)F*(H)的每个Sylow子群的所有极大子群在G中或者是π-拟正规的或者是c-可补的,其中F*(H)是H的广义Fitting子群.此结论统一了一些最近的结果.     

有限群的最大子群的性质对群结构的影响（英文）

有限群G的一个子群称为在G中是π-拟正规的若它与G的每一个Sylow-子群是交换的.G的一个子群H称为在G中是c-可补的若存在G的子群N使得G=HN且H∩N≤H_G=Core_G(H).本文证明了:设F是一个包含超可解群系U的饱和群系,G有一个正规子群H使得G/H∈F.则G∈F若下列之一成立:(1)H的每个Sylow子群的所有极大子群在G中或者是π-拟正规的或者是c-可补的;(2)F~*(H)的每个SyloW子群的所有极大子群在G中或者是π-拟正规的或者是c-可补的,其中F~*(H)是H的广义Fitting子群.此结论统一了一些最近的结果.     

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.     

Maximal Subgroups in the Classical Groups Normalizing Solvable Subgroups

Let G be a classical group over an arbitrary field F,acting on an n-dimensional F-space V=V(n,F).All those maximal subgroups of G are classified each of which normalizes a solvable subgroup N of GL(V/F)not lying in F*1v.     

Maximal Subgroups in the Classical Groups Normalizing Solvable Subgroups

Let G be a classical group over an arbitrary field F,acting on an n-dimensional vector space V=V(n,F) over a field F.In this paper,we classify the maximal subgroups of G,which normalizes a solvable subgroup N of GL(L,F) not lying in F~*1_V.