Centralizers in simple locally finite groups
Mahmut
Kuzucuoğlu
author
text
article
2013
eng
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 $\mathcal{K}=\{(G_i,N_i)\ |\ \ i\in \mathbf{N}\ \}$. If there exists an upper bound for $\{ |N_i| \ | \ \ i\in \mathbf{N}\ \}$, 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 $\{ |N_i| \ | \ \ i\in \mathbf{N}\ \}$
International Journal of Group Theory
University of Isfahan
2251-7650
2
v.
1
no.
2013
1
10
http://ijgt.ui.ac.ir/article_1521_f96e46ddc879945b9b1ada62f715c862.pdf
dx.doi.org/10.22108/ijgt.2013.1521
Replacement and zig-zag products, Cayley graphs and Lamplighter random walk
Alfredo
Donno
Università di Roma "La Sapienza"
author
text
article
2013
eng
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.
International Journal of Group Theory
University of Isfahan
2251-7650
2
v.
1
no.
2013
11
35
http://ijgt.ui.ac.ir/article_1932_cab6bdc876cc1f685e588bddab0243fc.pdf
dx.doi.org/10.22108/ijgt.2013.1932
Groups with all subgroups permutable or soluble
Martyn
Dixon
University of Alabama
author
Zekeriya
Karatas
University of Georgia
author
text
article
2013
eng
In this paper, we consider locally graded groups in which every non-permutable subgroup is soluble of bounded derived length.
International Journal of Group Theory
University of Isfahan
2251-7650
2
v.
1
no.
2013
37
43
http://ijgt.ui.ac.ir/article_2008_2258282ce9f951b8bdbf8f4d28f3ad05.pdf
dx.doi.org/10.22108/ijgt.2013.2008
Factorizing profinite groups into two abelian subgroups
Wolfgang
Herfort
University of Technology
author
text
article
2013
eng
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 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.
International Journal of Group Theory
University of Isfahan
2251-7650
2
v.
1
no.
2013
45
47
http://ijgt.ui.ac.ir/article_2341_935b81d6227dda83e4ad6d3d1bc07f37.pdf
dx.doi.org/10.22108/ijgt.2013.2341
Cocharacters of upper triangular matrices
Lucio
Centrone
Universita; degli Studi di Bari, II facolta; di scienze, Taranto
author
text
article
2013
eng
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.
International Journal of Group Theory
University of Isfahan
2251-7650
2
v.
1
no.
2013
49
77
http://ijgt.ui.ac.ir/article_2392_e062303d2c7de58331c72dbeafbb1519.pdf
dx.doi.org/10.22108/ijgt.2013.2392
Linear analogues of theorems of Schur, Baer and Hall
Martyn
Dixon
University
of Alabama
author
Leonid
Kurdachenko
National University of Dnepropetrovsk
author
Javier
Javier
University of Zaragoza
author
text
article
2013
eng
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.
International Journal of Group Theory
University of Isfahan
2251-7650
2
v.
1
no.
2013
79
89
http://ijgt.ui.ac.ir/article_2441_6e51a998a5edd8bad3956870b5f1843b.pdf
dx.doi.org/10.22108/ijgt.2013.2441
Certain combinatorial topics in group theory
C.
Gupta
University of Manitoba
author
text
article
2013
eng
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.
International Journal of Group Theory
University of Isfahan
2251-7650
2
v.
1
no.
2013
91
107
http://ijgt.ui.ac.ir/article_2439_e6c3c39c0bb1c0285110d472d9596943.pdf
dx.doi.org/10.22108/ijgt.2013.2439
On some invariants of finite groups
Jan
Krempa
Institute of Mathematics, University of Warsaw
author
Agnieszka
Stocka
Institute of Mathematics
University of Białystok
author
text
article
2013
eng
A normal subgroup $N$ of a group $G$ is said to be an omissible subgroup of $G$ if it has the following property: whenever $X\leq G$ is such that $G=XN$, then $G=X$. In this note we construct various groups $G$, each of which has an omissible subgroup $N\neq 1$ such that $G/N\cong SL_2(k)$ where $k$ is a field of positive characteristic.
International Journal of Group Theory
University of Isfahan
2251-7650
2
v.
1
no.
2013
109
115
http://ijgt.ui.ac.ir/article_2642_9f742c032856a8f0dcd5a8b7de56b25b.pdf
dx.doi.org/10.22108/ijgt.2013.2642
Metahamiltonian groups and related topics
Maria
De Falco
Dipartimento di Matematica e Applicazioni - University of Napoli "Federico II"
author
Francesco
de Giovanni
Dipartimento di Matematica e Applicazioni - University of Napoli "Federico II"
author
Carmela
Musella
Dipartimento di Matematica e Applicazioni - University of Napoli "Federico II"
author
text
article
2013
eng
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.
International Journal of Group Theory
University of Isfahan
2251-7650
2
v.
1
no.
2013
117
129
http://ijgt.ui.ac.ir/article_2673_d66ef73f5f08783c1a5e937dd4934f0c.pdf
dx.doi.org/10.22108/ijgt.2013.2673
Covering monolithic groups with proper subgroups
Martino
Garonzi
University of Padova
author
text
article
2013
eng
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 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.
International Journal of Group Theory
University of Isfahan
2251-7650
2
v.
1
no.
2013
131
144
http://ijgt.ui.ac.ir/article_2674_ea33a8a83df38a9fbe3ccb5bf483e473.pdf
dx.doi.org/10.22108/ijgt.2013.2674
Omissible extensions of $SL_2(k)$ where $k$ is a field of positive characteristic
Martyn
Dixon
The University of Alabama
author
Martin
Evans
The University of Alabama
author
Howard
Smith
Bucknell University
author
text
article
2013
eng
A normal subgroup $N$ of a group $G$ is said to be an omissible subgroup of $G$ if it has the following property: whenever $X\leq G$ is such that $G=XN$, then $G=X$. In this note we construct various groups $G$, each of which has an omissible subgroup $N\neq 1$ such that $G/N\cong SL_2(k)$ where $k$ is a field of positive characteristic.
International Journal of Group Theory
University of Isfahan
2251-7650
2
v.
1
no.
2013
145
155
http://ijgt.ui.ac.ir/article_2739_5cbd1f85252078e642252fe7f93e9285.pdf
dx.doi.org/10.22108/ijgt.2013.2739
Supersoluble conditions and transfer control
Anna Luisa
Gilotti
Universita Bologna
author
Luigi
Serena
Dipartimento di Matematica U. Dini
Viale Morgagni 67/A
50134 FIRENZE
author
text
article
2013
eng
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 characteri-zation 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 $p$-subgroup is given. These results make use of the properties of $G$-chains and $\Phi$-chains.
International Journal of Group Theory
University of Isfahan
2251-7650
2
v.
1
no.
2013
157
166
http://ijgt.ui.ac.ir/article_2740_448422e5d677e318f74f297e7784d5a0.pdf
dx.doi.org/10.22108/ijgt.2013.2740
A finiteness condition on the coefficients of the probabilistic zeta function
Hoang Dung
Duong
Mathematisch Instituut, Leiden Universiteit, Niels Bohrweg 1, 2333 CA Leiden, The Netherlands
author
Andrea
Lucchini
Dipartimento di Matematica
Università di Padova
author
text
article
2013
eng
We discuss whether finiteness properties of a profinite group $G$ can be deduced from the coefficients of the probabilistic zeta 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.
International Journal of Group Theory
University of Isfahan
2251-7650
2
v.
1
no.
2013
167
174
http://ijgt.ui.ac.ir/article_2760_6ca06204ff8803a9692e43ba9bbb64d6.pdf
dx.doi.org/10.22108/ijgt.2013.2760
The prime graph conjecture for integral group rings of some alternatings groups
Mohamed
Salim
UNITED ARAB EMIRATES UNIVERSITY
author
text
article
2013
eng
We investigate the classical H. Zassenhaus conjecture for integral group rings of alternating groups $A_9$ and $A_{10}$ of degree $9$ and $10$, respectively. As a consequence of our previous results we confirm the Prime Graph Conjecture for integral group rings of $A_n$ for all $n \leq 10$.
International Journal of Group Theory
University of Isfahan
2251-7650
2
v.
1
no.
2013
175
185
http://ijgt.ui.ac.ir/article_2759_c14fa9a571a2516ee5a9f8892f18fddc.pdf
dx.doi.org/10.22108/ijgt.2013.2759
On the representation theory of the alternating groups
Tullio
Ceccherini-Silberstein
Dipartimento di Ingegneria, Università del Sannio
author
Fabio
Scarabotti
Dipartimento SBAI, Sapienza Universita' di Roma
author
Filippo
Tolli
Dipartimento di Matematica e Fisica, Universita' Roma Tre
author
text
article
2013
eng
We present the basic results on the representation theory of the alternating groups $A_n$. Our approach is based on Clifford theory.
International Journal of Group Theory
University of Isfahan
2251-7650
2
v.
1
no.
2013
187
198
http://ijgt.ui.ac.ir/article_2841_21712515b0e5f7db680433e0809ea2ed.pdf
dx.doi.org/10.22108/ijgt.2013.2841
On finite arithmetic groups
Dmitry
Malinin
I.H.E.S.
author
text
article
2013
eng
Let $F$ be a finite extension of $\Bbb Q$, ${\Bbb Q}_p$ 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 $GL_n(E)$ 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 finite extension $E$ is realizable via adjoining to $F$ the entries of all matrices $g\in G$ 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 $G\subset GL_n(O_E)$ 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.
International Journal of Group Theory
University of Isfahan
2251-7650
2
v.
1
no.
2013
199
227
http://ijgt.ui.ac.ir/article_2865_15eedcf8206252b2004de346b12153d2.pdf
dx.doi.org/10.22108/ijgt.2013.2865