On subgroups containing non-trivial normal subgroups

We prove that ifA≠1 is a subgroup of a finite groupG and the order of an element in the centralizer ofA inG is strictly larger (larger or equal) than the index [G:A], thenA contains a non-trivial characteristic (normal) subgroup ofG. Consequently, ifA is a stabilizer in a transitive permutation group of degreem>1, thenexp(Z(A))<m. These theorems generalize some recent results of Isaacs and the authors.     

Explicit representations of maximal subgroups of the Monster

Most of the maximal subgroups of the Monster are now known, but in many cases they are hard to calculate in. We produce explicit ‘small’ representations of all the maximal subgroups which are not 2-local. The representations we construct are available on the World Wide Web at http://brauer.maths.qmul.ac.uk/Atlas/.     

A classification of subgroups of the Monster isomorphic to S4 and an application

We classify all subgroups of the Monster isomorphic to S₄. We then use this classification to prove that there are no maximal subgroups of the Monster with socles isomorphic to PSU₃(3), PSL₃(3), PSL₂(17), or PSL₂(7).     

A family of maximal subgroups containing the Sylow subgroups and some solvability conditions

Archiv der Mathematik -     

On discrete subgroups containing a lattice in a horospherical subgroup

We showed in [Oh] that for a simple real Lie groupG with real rank at least 2, if a discrete subgroup Γ ofG contains lattices in two opposite horospherical subgroups, then Γ must be a non-uniform arithmetic lattice inG, under some additional assumptions on the horospherical subgroups. Somewhat surprisingly, a similar result is true even if we only assume that Γ contains a lattice in one horospherical subgroup, provided Γ is Zariski dense inG.     

A Monster Graph I

The structure of the point-line collinearity graph of the maximal2-local geometry for the Monster simple group is investigated.The main results describe the first two discs around an arbitraryvertex. 2000 Mathematics Subject Classification 20D08.     

On the permutability of sylow subgroups with Schmidt subgroups

A Schmidt group is a finite nonnilpotent group in which every proper subgroup is nilpotent. Sufficient conditions are established for the p-solvability of a finite group in which a Sylow p-subgroup is permutable with some Schmidt subgroups. Sufficient conditions for the solvability of a finite group in which some Schmidt subgroups are permutable are also obtained.     

On the n-minimal subgroups

Let G be a finite group and n be a positive integer. An n-minimal subgroup H of G is called to be exactly n-minimal if no proper subgroup of H is n-minimal. In this paper, we study the solvability of G under the assumption that all exactly n-minimal subgroups of G are S-permutable.     

A Bruhat decomposition for subgroups containing the group of diagonal matrices. II

This paper is a continuation of RZhMat 1981, 7A438. Suppose R is a commutative ring generated by its group of units R^* and there exist such that. Suppose also that is the Jacobson radical of R, and B() is a subgroup of GL(n,R) consisting of the matrices a=(a_ij) such that a_ij for i>j. If a matrix aB() is represented in the form a=udv, where u is upper unitriangular, d is diagonal, and v is lower unitriangular, then u,vD,aDa^–1, where D=D(n,R) is the group of diagonal matrices. In particular, D is abnormal in B()Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 114, pp. 50–61, 1982.In conclusion, the author would like to thank Professor Z. I. Borevich for many useful discussions. The present paper (like [5, 12] and, in part [6, 7]) was mainly written in 1979–1980, while the author was at Wroclaw University. The author would like to thank his Wroclaw colleagues and, primarily, his adviser, Professor W. Narkiewicz, for creating ideal conditions under which to work.     

A Monster Lie algebra?

This paper defines a remarkable Lie algebra of infinite dimension and rank, and conjectures that it may be related to the Fischer-Griess Monster group.