2017
6
4
4
0
A new characterization of Ree group $mathbf{{}^2G_2(q)}$ by the order of group and the number of elements with the same order
2
2
In this paper, we prove that Ree group ${}^2G_2(q)$, where $qpmsqrt{3q}+1$ is a prime number can be uniquely determined by the order of group and the number of elements with the same order.
1

1
6


Behnam
Ebrahimzadeh
‎Islamic Azad University
‎Islamic Azad University
Iran
behnam.ebrahimzadeh@gmail.com


Ali
Iranmanesh
Tarbiat Modares University
Tarbiat Modares University
Iran
iranmana@modares.ac.ir


Hosein
Parvizi Mosaed
Alvand Institute of Higher Education
Alvand Institute of Higher Education
Iran
h.parvizi.mosaed@gmail.com
Element order
Prime graph
Ree group
On almost recognizability by spectrum of simple classical groups
2
2
The set of element orders of a finite group $G$ is called the {em spectrum}. Groups with coinciding spectra are said to be {em isospectral}. It is known that if $G$ has a nontrivial normal soluble subgroup then there exist infinitely many pairwise nonisomorphic groups isospectral to $G$. The situation is quite different if $G$ is a nonabelain simple group. Recently it was proved that if $L$ is a simple classical group of dimension at least 62 and $G$ is a finite group isospectral to $L$, then up to isomorphism $Lleq GleqAut L$. We show that the assertion remains true if 62 is replaced by 38.
1

7
33


Alexey
Staroletov
Iran
astaroletov@gmail.com
Simple classical groups
Element orders
Prime graph of a finite group
Almost recognizable group
Finite nonnilpotent groups with few nonnormal noncyclic subgroups
2
2
For a finite group $G$, let $nu_{nc}(G)$ denote the number of conjugacy classes of nonnormal noncyclic subgroups of $G$. We characterize the finite nonnilpotent groups whose all nonnormal noncyclic subgroups are conjugate.
1

35
40


Hamid
Mousavi
Department of Mathematical Sciences, University of Tabriz, P.O.Box 5166616471, Tabriz, Iran
Department of Mathematical Sciences,
Iran
hmousavi@tabrizu.ac.ir


Zahra
Rezazadeh
Department of Mathematical Sciences, Isfahan University of Technology, P.O.Box 8415683111, Isfahan, Iran
Department of Mathematical Sciences
Iran
zahra.rezazadeh@math.iut.ac.ir
Nonnormal subgroups
Conjugacy classes of nonnormal subgroups
Nonnilpotent groups
Bipartite divisor graph for the set of irreducible character degrees
2
2
Let $G$ be a finite group. We consider the set of the irreducible complex characters of $G$, namely $Irr(G)$, and the related degree set $cd(G)={chi(1) : chiin Irr(G)}$. Let $rho(G)$ be the set of all primes which divide some character degree of $G$. In this paper we introduce the bipartite divisor graph for $cd(G)$ as an undirected bipartite graph with vertex set $rho(G)cup (cd(G)setminus{1})$, such that an element $p$ of $rho(G)$ is adjacent to an element $m$ of $cd(G)setminus{1}$ if and only if $p$ divides $m$. We denote this graph simply by $B(G)$. Then by means of combinatorial properties of this graph, we discuss the structure of the group $G$. In particular, we consider the cases where $B(G)$ is a path or a cycle.
1

41
51


Roghayeh
Hafezieh
GEBZE TECHNICAL UNIV.
GEBZE TECHNICAL UNIV.
Turkey
roghayeh@gtu.edu.tr
bipartite divisor graph
irreducible character degree
path
cycle
The conjugacy class ranks of $M_{24}$
2
2
$M_{24}$ is the largest Mathieu sporadic simple group of order $244 823 040 = 2^{10} {cdot} 3^3 {cdot} 5 {cdot} 7 {cdot} 11 {cdot} 23$ and contains all the other Mathieu sporadic simple groups as subgroups. The object in this paper is to study the ranks of $M_{24}$ with respect to the conjugacy classes of all its nonidentity elements.
1

53
58


Zwelethemba
Mpono
University of South Africa
University of South Africa
South Africa
mponoze@unisa.ac.za
Rank
Generations
structure constants
class fusions
maximal subgroups