INTERNATIONAL JOURNAL OF GROUP THEORY
Year:2013 | Volume:2 | Issue:1|Papers Count:16

This is a survey article on centralizers of finite subgroups in locally finite, simple groups or LFS-groups as we will call them. We mention some of the open problems about centralizers of subgroups in LFS-groups and applications of the known information about the centralizers of subgroups to the structure of the locally finite group. We also prove the following: Let G be a countably in finite nonlinear LFS-group with a Kegel sequence K={(Gi, Ni) | iÎN}. If there exists an upper bound for {|Ni| | iÎN}, then for any finite semisimple subgroup F in G the subgroup CG (F) has elements of order pi for in finitely many distinct prime pi. In particular CG (F) is an in finite group. This answers Hartley’s question provided that there exists a bound on {|Ni| | iÎN}.

We investigate two constructions -the replacement and the zig-zag product of graphs- describing several fascinating connections with Combinatorics, via the notion of expander graph, Group Theory, via the notion of semidirect product and Cayley graph, and with Markov chains, via the Lamplighter random walk. Many examples are provided.

In this paper, we consider locally graded groups in which every non-permutable subgroup is soluble of bounded derived length.

We prove that the class of pro finite groups G that have a factorization G=AB with A and Babelian closed subgroups, is closed under taking inverse limits of surjective inverse systems. This is a generalization of a recent result by K.H. Hofmann and F.G. Russo. As an application we reprove their generalization of Iwasawa’s structure theorem for quasihamiltonian pro-p groups.

We survey some recent results on cocharacters of upper triangular matrices. In particular, we deal both with ordinary and graded cocharacter sequence; we list the principal combinatorial results; we show different techniques in order to solve similar problems.

A celebrated result of I. Schur asserts that the derived subgroup of a group is finite provided the group modulo its center is finite, a result that has been the source of many investigations within the Theory of Groups. In this paper we exhibit a similar result to Schur’s Theorem for vector spaces, acted upon by certain groups. The proof of this analogous result depends on the characteristic of the underlying field. We also give linear versions of corresponding theorems of R. Baer and P. Hall.

This article is intended to be a survey on some combinatorial topics in group theory. The bibliography at the end is neither claimed to be exhaustive, nor is it necessarily connected with a reference in the text. I include it as I see it revolves around the concepts which are discussed in the text.

In this note we are going to survey several invariants of finite groups related either to their orders or to generating sets or to lattices of subgroups. Some relations among these invariants will be exhibited. Special attention will be paid to monotonicity of them.

A group G is called metahamiltonian if all its non-abelian subgroups are normal. The aim of this paper is to provide an updated survey of research concerning certain classes of generalized metahamiltonian groups, in various contexts, and to prove some new results. Some open problems are listed.

Given a finite non-cyclic group G, call s (G) the smallest number of proper subgroups of G needed to cover G. Lucchini and Detomi conjectured that if a nonabelian group G is such that s (G)<s (G/N) for every non-trivial normal subgroup N of G then G is monolithic, meaning that it admits a unique minimal normal subgroup. In this paper we show how this conjecture can be attacked by the direct study of monolithic groups.

A normal subgroup N of a group G is said to be an omissible subgroup of G if it has the following property: whenever X£G is such that G=XN, then G=X. In this note we construct various groups G, each of which has an omissible subgroup N¹1 such that G/N@SL2(k) where k is a field of positive characteristic.

In this paper we give a new condition for a Sylow r-subgroup of a finite group to control transfer. Then it is deduced a characterization of supersoluble groups that can be seen as a generalization of the well-known result concerning the supersolubility of finite groups with cyclic Sylow subgroups. Moreover a condition for a normal embedding of a strongly closed r-subgroup is given. These results make use of the properties of G-chains and F-chains.

We discuss whether finiteness properties of a pro finite group G can be deduced from the coefficients of the probabilistic zeta function PG(s). In particular we prove that if PG(s) is rational and all but finitely many non abelian composition factors of G are isomorphic to PSL (2, p) for some prime p, then G contains only finitely many maximal subgroups.

We investigate the classical H. Zassenhaus conjecture for integral group rings of alternating groups A9 and A10 of degree 9 and 10, respectively. As a consequence of our previous results we con rm the Prime Graph Conjecture for integral group rings of An for all n£10.

We present the basic results on the representation theory of the alternating groups ‚n. Our approach is based on Clifford theory.

Let F be a finite extension of Q, Qp or a global field of positive characteristic, and let E/F be a Galois extension. We study the realization fields of finite subgroups G of GLn(E) stable under the natural operation of the Galois group of E/F. Though for sufficiently large n and a fixed algebraic number field F every its nite extension E is realizable via adjoining to F the entries of all matrices gÎG for some finite Galois stable subgroup G of GLn(C), there is only a finite number of possible realization field extensions of F if GÌGLn(OE) over the ring OE of integers of E. After an exposition of earlier results we give their refinements for the realization fields E/F. We consider some applications to quadratic lattices, arithmetic algebraic geometry and Galois cohomology of related arithmetic groups.