Centralizers in a group whose central factor is simple

2018 ◽  
Vol 17 (08) ◽  
pp. 1850149
Author(s):  
Seyyed Majid Jafarian Amiri ◽  
Hojjat Rostami
Keyword(s):  
Finite Group ◽  
Linear Group ◽  
Finite Groups ◽  
Special Linear Group ◽  
Projective Special Linear Group ◽  
Special Linear ◽  
Central Factor ◽  
Suzuki Group ◽  
A Finite Group ◽  
Appl Math

In this paper, we find the number of the element centralizers of a finite group [Formula: see text] such that the central factor of [Formula: see text] is the projective special linear group of degree 2 or the Suzuki group. Our results generalize some main results of [Ashrafi and Taeri, On finite groups with a certain number of centralizers, J. Appl. Math. Comput. 17 (2005) 217–227; Schmidt, Zentralisatorverbände endlicher Gruppen, Rend. Sem. Mat. Univ. Padova 44 (1970) 97–131; Zarrin, On element centralizers in finite groups, Arch. Math. 93 (2009) 497–503]. Also, we give an application of these results.

scholarly journals On recognition by prime graph of the projective special linear group over GF(3)

2014 ◽  
Vol 95 (109) ◽  
pp. 255-266
Author(s):  
Bahman Khosravi ◽  
Behnam Khosravi ◽  
Oskouei Dalili
Keyword(s):  
Finite Group ◽  
Simple Group ◽  
Linear Group ◽  
Prime Graph ◽  
Special Linear Group ◽  
Composition Factor ◽  
Projective Special Linear Group ◽  
Special Linear ◽  
A Finite Group ◽  
Nonabelian Composition Factor

Let G be a finite group. The prime graph of G is denoted by ?(G). We prove that the simple group PSLn(3), where n ? 9, is quasirecognizable by prime graph; i.e., if G is a finite group such that ?(G) = ?(PSLn(3)), then G has a unique nonabelian composition factor isomorphic to PSLn(3). Darafsheh proved in 2010 that if p > 3 is a prime number, then the projective special linear group PSLp(3) is at most 2-recognizable by spectrum. As a consequence of our result we prove that if n ? 9, then PSLn(3) is at most 2-recognizable by spectrum.

scholarly journals Families of solvable Frobenius subgroups in finite groups

2002 ◽  
Vol 165 ◽  
pp. 117-121
Author(s):  
Paul Lescot
Keyword(s):  
Finite Group ◽  
Linear Group ◽  
Finite Groups ◽  
Special Linear Group ◽  
Famous Result ◽  
Rank 2 ◽  
Characteristic 2 ◽  
A Finite Group ◽  
Suzuki Groups ◽  
Degenerate Cases

We introduce the notion of abelian system on a finite group G, as a particular case of the recently defined notion of kernel system (see this Journal, September 2001). Using a famous result of Suzuki on CN-groups, we determine all finite groups with abelian systems. Except for some degenerate cases, they turn out to be special linear group of rank 2 over fields of characteristic 2 or Suzuki groups. Our ideas were heavily influenced by [1] and [8].

scholarly journals Computing of Z- valued Characters for the Projective Special Linear Group L2 (2m) and the Conway Group Co3

2016 ◽  
Vol 8 (3) ◽  
pp. 61 ◽  
Author(s):  
Ali Moghani
Keyword(s):  
Group Theory ◽  
Nonlinear Optics ◽  
Linear Group ◽  
Finite Groups ◽  
Special Linear Group ◽  
Projective Special Linear Group ◽  
Special Linear ◽  
Dominant Classes

<p><span lang="EN-US">According to the main result of W. Feit and G. M. Seitz (see, Illinois J. Math. 33 (1), 103-131, 1988), the projective special linear group L<sub>2</sub> (2<sup>m</sup>) for m = 3, 4, 5 and the smallest Conway group Co<sub>3</sub> are unmatured groups. In this paper, we continue our study on special finite groups (see Int. J. Theo. Physics, Group Theory, and Nonlinear Optics (17)1, 57-62, 2013) and the dominant classes and Q- conjugacy characters for the above groups are derived.</span></p>

scholarly journals Finite automorphism groups of laminated near-rings

1983 ◽  
Vol 26 (3) ◽  
pp. 297-306 ◽  
Author(s):  
K. D. Magill ◽  
P. R. Misra ◽  
U. B. Tewari
Keyword(s):  
Finite Group ◽  
Linear Group ◽  
Complex Plane ◽  
Finite Groups ◽  
Automorphism Groups ◽  
The Other ◽  
Complex Polynomial ◽  
Infinite Groups ◽  
Nonsingular Matrices ◽  
A Finite Group

In [3] we initiated our study of the automorphism groups of a certain class of near-rings. Specifically, let P be any complex polynomial and let P denote the near-ring of all continuous selfmaps of the complex plane where addition of functions is pointwise and the product fg of two functions f and g in P is defined by fg=f∘P∘g. The near-ring P is referred to as a laminated near-ring with laminating element P. In [3], we characterised those polynomials P(z)=anzn + an−1zn−1 +…+a0 for which Aut P is a finite group. We are able to show that Aut P is finite if and only if Deg P≧3 and ai ≠ 0 for some i ≠ 0, n. In addition, we were able to completely determine those infinite groups which occur as automorphism groups of the near-rings P. There are exactly three of them. One is GL(2) the full linear group of all real 2×2 nonsingular matrices and the other two are subgroups of GL(2). In this paper, we begin our study of the finite automorphism groups of the near-rings P. We get a result which, in contrast to the situation for the infinite automorphism groups, shows that infinitely many finite groups occur as automorphism groups of the near-rings under consideration. In addition to this and other results, we completely determine Aut P when the coefficients of P are real and Deg P = 3 or 4.

scholarly journals Cyclotomic equations and square properties in rings

1986 ◽  
Vol 9 (1) ◽  
pp. 89-95
Author(s):  
Benjamin Fine
Keyword(s):  
Linear Group ◽  
Special Linear Group ◽  
Sum Of Squares ◽  
Polynomial Rings ◽  
Projective Special Linear Group ◽  
Infinite Class ◽  
Related Property ◽  
Special Linear

IfRis a ring, the structure of the projective special linear groupPSL2(R)is used to investigate the existence of sum of square properties holding inR. Rings which satisfy Fermat's two-square theorem are called sum of squares rings and have been studied previously. The present study considers a related property called square property one. It is shown that this holds in an infinite class of rings which includes the integers, polynomial rings over many fields andZpnwherePis a prime such that−3is not a squaremodp. Finally, it is shown that the class of sum of squares rings and the class satisfying square property one are non-coincidental.

RECOGNIZING BY ORDER AND DEGREE PATTERN OF SOME PROJECTIVE SPECIAL LINEAR GROUPS

2012 ◽  
Vol 22 (06) ◽  
pp. 1250051 ◽  
Author(s):  
B. AKBARI ◽  
A. R. MOGHADDAMFAR
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Linear Groups ◽  
Special Linear ◽  
Projective Special Linear Groups ◽  
Special Linear Groups ◽  
Isomorphism Classes ◽  
Degree Pattern ◽  
A Finite Group

Let M be a finite group and D (M) be the degree pattern of M. Denote by h OD (M) the number of isomorphism classes of finite groups G with the same order and degree pattern as M. A finite group M is called k-fold OD-characterizable if h OD (M) = k. Usually, a 1-fold OD-characterizable group is simply called OD-characterizable. The purpose of this article is twofold. First, it provides some information on the structure of a group from its degree pattern. Second, it proves that the projective special linear groups L4(4), L4(8), L4(9), L4(11), L4(13), L4(16), L4(17) are OD-characterizable.

scholarly journals De non ordinaria recognitione PSL2

2008 ◽  
Vol 7 (4) ◽  
pp. 723-734
Author(s):  
Adrien Deloro
Keyword(s):  
Linear Group ◽  
Special Linear Group ◽  
Morley Rank ◽  
Projective Special Linear Group ◽  
Special Linear ◽  
Dimension 2

AbstractWe establish an identification result of the projective special linear group of dimension 2 among a certain class of groups the Morley rank of which is finite.

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