فهرست مطالب

International Journal of Group Theory - Volume:2 Issue:1, 2012
  • Volume:2 Issue:1, 2012
  • تاریخ انتشار: 1391/08/15
  • تعداد عناوین: 16
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  • Mahmut Kuzucuoglu Pages 1-10
    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 infinite non-linear LFS-group with a Kegel sequence $mathcal{K}={(G_i,N_i) | iin mathbf{N}}$. If there exists an upper bound for ${|N_i| | iin mathbf{N}}$, then for any finite semisimple subgroup $F$ in $G$ the subgroup $C_G(F)$ has elements of order $p_i$ for infinitely many distinct prime $p_i$. In particular $C_G(F)$ is an infinite group. This answers Hartley''s question provided that there exists a bound on ${|N_i| | iin mathbf{N}}$
    Keywords: Centralizer, locally finite, simple groups
  • Alfredo Donno Pages 11-35
    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.
    Keywords: Replacement, zig, zag product, Expander graph, Lamplighter random walk, Cayley graph, Semidirect, wreath product
  • Martyn Dixon, Zekeriya Karatas Pages 37-43
    In this paper, we consider locally graded groups in which every non-permutable subgroup is soluble of bounded derived length.
    Keywords: permutable, soluble, locally graded
  • Wolfgang Herfort Pages 45-47
    We prove that the class of profinite groups $G$ that have a factorization $G=AB$ with $A$ and $B$ abelian closed subgroups, is closed under taking strict projective limits. 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.
    Keywords: group factorization, pro, $p$ groups, limits
  • Lucio Centrone Pages 49-77
    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 di erent tech-niques in order to solve similar problems.
    Keywords: Upper triangular matrices, cocharacters, Grassmann algebra
  • Martyn Dixon, Leonid Kurdachenko, Javier Javier Pages 79-89
    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.
    Keywords: derived submodule of a module over a group ring, section $p$, rank, section $0$, rank, linear group
  • C. K Gupta Pages 91-107
    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.
    Keywords: automorphisms, test words, polynilpotent series, universal theories
  • Jan Krempa, Agnieszka Stocka Pages 109-115
    In this note we are going to survey several invariants of finite groups related either to theirorders 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.
    Keywords: generating set, independent set, (p, q), group, lattice of subgroups
  • Maria De Falco, Francesco De Giovanni, Carmela Musella Pages 117-129
    A group is called metahamiltonian if all its non-abelian subgroups are normal. This aim of this paper is to provide an updated survey of researches concerning certain classes of generalized metahamiltonian groups, in various contexts, and to prove some new results. Some open problems are listed.
    Keywords: Metahamiltonian group, normalizer subgroup, lattice property
  • Martino Garonzi Pages 131-144
    Given a finite non-cyclic group $G$, call $sigma(G)$ 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 $sigma(G) < sigma(G/N)$ for every non-trivial normal subgroup $N$ of $G$ then $G$ is textit{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.
    Keywords: Covers, Monolithic groups, Primitive groups
  • Martyn Dixon, Martin Evans, Howard Smith Pages 145-155
    A normal subgroup $N$ of a group $G$ is said to be an$emph{omissible}$ subgroup of $G$ if it has the following property: whenever $Xleq G$ is such that $G=XN$, then $G=X$.In this note we construct various groups $G$, each of which has an omissible subgroup $Nneq 1$ such that $G/Ncong SL_2(k)$ where $k$ is a field of positive characteristic.
    Keywords: Omissible subgroup, special linear group, Frattini extension, locally (soluble, by, finite) group
  • Anna Luisa Gilotti, Luigi Serena Pages 157-166
    In this paper we give a new condition for a Sylow p-subgroup of a finite group to control transfer. Then it is deduced a characterization of supersoluble group that can be seen as a generalization of the well konwn result concerning the supersolubility of finite groups with cyclic Sylow subgroups. Moreover a condition for a normal embedding of a strongly closed p-subgroup is given These results make use of the properties of G-chain and Phi-chain.
    Keywords: transfer control, normal p, complement, supersoluble groups, strongly closed subgroup
  • Hoang Dung Duong, Andrea Lucchini Pages 167-174
    We discuss whether finiteness properties of a profinite group $G$ can be deduced from the coefficients of the probabilisticzeta function $P_G(s)$. In particular we prove that if $P_G(s)$ 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.
    Keywords: Probabilistic zeta function, finiteness conditions, special linear groups
  • Mohamed Salim Pages 175-185
    We investigate the classical Zassenhaus Conjecture (ZC) for integral group rings of alternating groups A9 and A10. Even the question (ZC) remains open as no counterexample is known up to date, it been confirmed for special types of groups such as nilpotent groups by Roggenkamp, Scot and Weiss. However, a new method based on the partial augmentation of torsion units been established by Luthar and Passi to confirm the (ZC) for A5. Later a weaker version of (ZC) was proposed in 2007, we call it the Prime Graph Conjecture (PGC) about the Gruenberg-Kegel (prime) graph of the group of all normalized units of the integral group ring of a finite group. Recently, the (PGC) has a positive answer for solvable groups, Frobenius groups and several simple groups. Here, as a consequence of our results, we confirm the (PGC) for integral group rings of alternating groups An for all n<11.
    Keywords: Integral group ring, Zassenhaus Conjecture, Prime Graph Conjecture, torsion unit
  • Tullio Ceccherini, Silberstein, Fabio Scarabotti, Filippo Tolli Pages 187-198
    We present the basic results on the representation theory of the alternating groups. Our approach is based on Clifford theory.
    Keywords: alternating group, irreducible representation, character, Conjugacy Class, Clifford Theory
  • Dmitry Malinin Pages 199-227
    begin{abstract} Let $F$ be a finite extension of $Bbb Q$, ${Bbb Q}_p$ or a globalfield of positive characteristic, and let $E/F$ be a Galois extension.We study the realization fields of finite subgroups $G$ of $GL_n(E)$ stable under the naturaloperation of the Galois group of $E/F$. Though for sufficiently large $n$ and a fixed algebraic number field $F$ every its finite extension $E$ is realizable via adjoining to $F$ the entries of all matrices $gin G$ for some finite Galois stable subgroup $G$ of $GL_n(Bbb C)$, there is only a finite number of possible realization field extensions of $F$ if $Gsubset GL_n(O_E)$ over the ring $O_E$ 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.
    Keywords: algebraic integers, Galois groups, integral representations, realization fields