Recognition of 2-dimensional projective linear groups by the group order and the set of numbers of its elements of each order

2018 ◽  
Vol 0 (0) ◽  
Author(s):  
Alireza Khalili Asboei
Keyword(s):  
Finite Group ◽  
Prime Power ◽  
Linear Groups ◽  
Group Order ◽  
A Finite Group ◽  
Projective Linear Groups ◽  
Orders Of Elements

Abstract In a finite group G, let {\pi_{e}(G)} be the set of orders of elements of G, let {s_{k}} denote the number of elements of order k in G, for each {k\in\pi_{e}(G)} , and then let {\operatorname{nse}(G)} be the unordered set {\{s_{k}:k\in\pi_{e}(G)\}} . In this paper, it is shown that if {\lvert G\rvert=\lvert L_{2}(q)\rvert} and {\operatorname{nse}(G)=\operatorname{nse}(L_{2}(q))} for some prime-power q, then G is isomorphic to {L_{2}(q)} .

A New Characterization of PSL(2, q) by Group Order and the Set of Vanishing Element Orders

2019 ◽  
Vol 26 (03) ◽  
pp. 459-466
Author(s):  
Changguo Shao ◽  
Qinhui Jiang
Keyword(s):  
Finite Group ◽  
Prime Power ◽  
Complex Character ◽  
Element Orders ◽  
Group Order ◽  
Vanishing Elements ◽  
A Finite Group

An element g in a finite group G is called a vanishing element if there exists some irreducible complex character χ of G such that [Formula: see text]. Denote by Vo(G) the set of orders of vanishing elements of G, and we prove that [Formula: see text] if and only if [Formula: see text] and [Formula: see text], where [Formula: see text] is a prime power.

On S-Quasinormal and C-Normal Subgroups of Prime Power Order in Finite Groups

2011 ◽  
Vol 18 (04) ◽  
pp. 685-692
Author(s):  
Xuanli He ◽  
Shirong Li ◽  
Xiaochun Liu
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Prime Power ◽  
Maximal Subgroups ◽  
Prime Power Order ◽  
Normal Subgroups ◽  
A Finite Group

Let G be a finite group, p the smallest prime dividing the order of G, and P a Sylow p-subgroup of G with the smallest generator number d. Consider a set [Formula: see text] of maximal subgroups of P such that [Formula: see text]. It is shown that if every member [Formula: see text] of is either S-quasinormally embedded or C-normal in G, then G is p-nilpotent. As its applications, some further results are obtained.

scholarly journals Automorphism orbits of finite groups

1986 ◽  
Vol 40 (2) ◽  
pp. 253-260 ◽  
Author(s):  
Thomas J. Laffey ◽  
Desmond MacHale
Keyword(s):  
Finite Group ◽  
Automorphism Group ◽  
Finite Groups ◽  
Prime Power ◽  
Structure Theorem ◽  
Equivalence Classes ◽  
Prime Power Order ◽  
A Finite Group

AbstractLet G be a finite group and let Aut(G) be its automorphism group. Then G is called a k-orbit group if G has k orbits (equivalence classes) under the action of Aut(G). (For g, hG, we have g ~ h if ga = h for some Aut(G).) It is shown that if G is a k-orbit group, then kGp + 1, where p is the least prime dividing the order of G. The 3-orbit groups which are not of prime-power order are classified. It is shown that A5 is the only insoluble 4-orbit group, and a structure theorem is proved about soluble 4-orbit groups.

scholarly journals On the nilpotency of the solvable radical of a finite group isospectral to a simple group

2020 ◽  
Vol 23 (3) ◽  
pp. 447-470
Author(s):  
Nanying Yang ◽  
Mariya A. Grechkoseeva ◽  
Andrey V. Vasil’ev
Keyword(s):  
Finite Group ◽  
Simple Group ◽  
Finite Simple Group ◽  
Solvable Radical ◽  
A Finite Group ◽  
Orders Of Elements

AbstractWe refer to the set of the orders of elements of a finite group as its spectrum and say that groups are isospectral if their spectra coincide. We prove that, except for one specific case, the solvable radical of a nonsolvable finite group isospectral to a finite simple group is nilpotent.

ON SPECTRUM OF LINEAR GROUPS OVER THE BINARY FIELD AND RECOGNIZABILITY OF L12(2)

2006 ◽  
Vol 16 (02) ◽  
pp. 341-349 ◽  
Author(s):  
A. R. MOGHADDAMFAR
Keyword(s):  
Finite Group ◽  
Computer Program ◽  
Simple Group ◽  
Linear Groups ◽  
Element Orders ◽  
Binary Field ◽  
Set Of Element Orders ◽  
A Finite Group

The spectrum ω(G) of a finite group G is the set of element orders of G. A finite group G is said to be recognizable through its spectrum, if for every finite group H, the equality of the spectra ω(H) = ω(G) implies the isomorphism H ≅ G. In this paper, first we try to write a computer program for computing ω(Ln(2)) for any n ≥ 3. Then, we will show that the simple group L12(2) is recognizable through its spectrum.

Recognition of Finite Simple Linear Groups L4(2k) by Spectrum

2010 ◽  
Vol 17 (03) ◽  
pp. 469-474
Author(s):  
Mingchun Xu
Keyword(s):  
Finite Group ◽  
Linear Groups ◽  
Element Orders ◽  
Set Of Element Orders ◽  
A Finite Group

A finite group G is said to be recognizable by spectrum, i.e., by the set of element orders, if every finite group H having the same spectrum as G is isomorphic to G. Grechkoseeva, Shi and Vasilev have proved that the simple linear groups Ln(2k) are recognizable by spectrum for n=2m≥ 16. In this paper we establish the recognizability for the case n=4.

The Influence of Certain Abelian Subgroups on the Structure of Finite Groups

2008 ◽  
Vol 15 (03) ◽  
pp. 479-484 ◽  
Author(s):  
M. Ramadan
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Prime Power ◽  
Prime Power Order ◽  
Group A ◽  
Abelian Subgroups ◽  
A Finite Group

Let G be a finite group. A subgroup K of a group G is called an [Formula: see text]-subgroup of G if NG(K) ∩ Kx ≤ K for all x ∈ G. The set of all [Formula: see text]-subgroups of G is denoted by [Formula: see text]. In this paper, we investigate the structure of a group G under the assumption that certain abelian subgroups of prime power order belong to [Formula: see text].

scholarly journals Recognition of symplectic and orthogonal groups of small dimensions by spectrum

2019 ◽  
Vol 18 (12) ◽  
pp. 1950230
Author(s):  
Mariya A. Grechkoseeva ◽  
Andrey V. Vasil’ev ◽  
Mariya A. Zvezdina
Keyword(s):  
Finite Group ◽  
Simple Group ◽  
Finite Groups ◽  
Simple Groups ◽  
Orthogonal Groups ◽  
A Finite Group ◽  
Field Automorphism ◽  
Orders Of Elements

We refer to the set of the orders of elements of a finite group as its spectrum and say that finite groups are isospectral if their spectra coincide. In this paper, we determine all finite groups isospectral to the simple groups [Formula: see text], [Formula: see text], and [Formula: see text]. In particular, we prove that with just four exceptions, every such finite group is an extension of the initial simple group by a (possibly trivial) field automorphism.

Finite Linear Groups of Degree Seven. I

1969 ◽  
Vol 21 ◽  
pp. 1042-1053 ◽  
Author(s):  
David B. Wales
Keyword(s):  
Finite Group ◽  
Complex Representation ◽  
Linear Groups ◽  
Subsequent Paper ◽  
Prime Degree ◽  
Sylow Group ◽  
A Finite Group ◽  
Finite Linear

1. 1. This paper is the second in a series of papers discussing linear groups of prime degree, the first being (8). In this paper we discuss only linear groups of degree 7. Thus, G is a finite group with a faithful irreducible complex representation Xof degree 7 which is unimodular and primitive. The character of Xis x- The notation of (8) is used except here p= 7. Thus Pis a 7-Sylow group of G.In §§ 2 and 3 some general theorems about the 3-Sylow group and 5-Sylow group are given. In § 4 the statement of the results when Ghas a non-abelian 7-Sylow group is given. This corresponds to the case |P| =73 or |P|= 74. The proof is given in §§ 5 and 6. In a subsequent paper the results when Pis abelian will be given.

scholarly journals Lie powers of free modules for certain groups of prime power order

2001 ◽  
Vol 71 (2) ◽  
pp. 149-158 ◽  
Author(s):  
R. M. Bryant ◽  
I. C. Michos
Keyword(s):  
Finite Group ◽  
Prime Power ◽  
Prime Power Order ◽  
Finite Dimensional ◽  
A Finite Group ◽  
Free Modules

AbstractLet G be a finite group of order pk, where p is a prime and k ≥ 1, such that G is either cyclic, quaternion or generalised quaternion. Let V be a finite-dimensional free KG-module where K is a field of characteristic p. The Lie powers Ln(V) are naturally KG-modules and the main result identifies these modules up to isomorphism. There are only two isomorphism types of indecomposables occurring as direct summands of these modules, namely the regular KG-module and the indecomposable of dimension pk – pk−1 induced from the indecomposable K H-module of dimension p − 1, where H is the unique subgroup of G of order p. Formulae are given for the multiplicities of these indecomposables in Ln(V). This extends and utilises work of the first author and R. Stöhr concerned with the case where G has order p.

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