scholarly journals Odd order products of conjugate involutions in~linear groups over GF(2𝑎)

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
John J. Ballantyne ◽  
Peter J. Rowley
Keyword(s):  
Conjugacy Class ◽  
Linear Groups ◽  
Odd Order

Abstract Let 𝐺 be isomorphic to GL n ⁢ ( q ) \mathrm{GL}_{n}(q) , SL n ⁢ ( q ) \mathrm{SL}_{n}(q) , PGL n ⁢ ( q ) \mathrm{PGL}_{n}(q) or PSL n ⁢ ( q ) \mathrm{PSL}_{n}(q) , where q = 2 a q=2^{a} . If 𝑡 is an involution lying in a 𝐺-conjugacy class 𝑋, then, for arbitrary 𝑛, we show that, as 𝑞 becomes large, the proportion of elements of 𝑋 which have odd order product with 𝑡 tends to 1. Furthermore, for 𝑛 at most 4, we give formulae for the number of elements in 𝑋 which have odd order product with 𝑡, in terms of 𝑞.

Nilpotent metacyclic irreducible linear groups of odd order

1995 ◽  
Vol 65 (4) ◽  
pp. 281-288 ◽  
Author(s):  
L. G. Kov�cs ◽  
Hyo -Seob Sim
Keyword(s):  
Linear Groups ◽  
Odd Order

scholarly journals On solvable groups with one vanishing class size

2020 ◽  
pp. 1-20
Author(s):  
M. Bianchi ◽  
E. Pacifici ◽  
R. D. Camina ◽  
Mark L. Lewis
Keyword(s):  
Conjugacy Class ◽  
Finite Groups ◽  
Class Size ◽  
Single Element ◽  
Conjugacy Class Sizes ◽  
Conjugacy Class Size ◽  
Odd Order ◽  
Vanishing Elements ◽  
A Finite Group

Let G be a finite group, and let cs(G) be the set of conjugacy class sizes of G. Recalling that an element g of G is called a vanishing element if there exists an irreducible character of G taking the value 0 on g, we consider one particular subset of cs(G), namely, the set vcs(G) whose elements are the conjugacy class sizes of the vanishing elements of G. Motivated by the results inBianchi et al. (2020, J. Group Theory, 23, 79–83), we describe the class of the finite groups G such that vcs(G) consists of a single element under the assumption that G is supersolvable or G has a normal Sylow 2-subgroup (in particular, groups of odd order are covered). As a particular case, we also get a characterization of finite groups having a single vanishing conjugacy class size which is either a prime power or square-free.

scholarly journals On the odd order composition factors of finite linear groups

2020 ◽  
Vol 23 (6) ◽  
pp. 1057-1068
Author(s):  
Alexander Betz ◽  
Max Chao-Haft ◽  
Ting Gong ◽  
Anthony Ter-Saakov ◽  
Yong Yang
Keyword(s):  
Linear Group ◽  
Linear Groups ◽  
Composition Series ◽  
Odd Order ◽  
Finite Linear Group ◽  
Composition Factors ◽  
Finite Linear

AbstractIn this paper, we study the product of orders of composition factors of odd order in a composition series of a finite linear group. First we generalize a result by Manz and Wolf about the order of solvable linear groups of odd order. Then we use this result to find bounds for the product of orders of composition factors of odd order in a composition series of a finite linear group.

scholarly journals On Irreducible Linear Groups of Nonprimary Degree

ISRN Algebra ◽  
2011 ◽  
Vol 2011 ◽  
pp. 1-20 ◽  
Author(s):  
A. A. Yadchenko
Keyword(s):  
Prime Number ◽  
Linear Group ◽  
Prime Power ◽  
Linear Groups ◽  
Odd Order ◽  
Complex Linear

It is established that degree 2|A|+1 of irreducible complex linear group with the group A of cosimple automorphisms of odd order is a prime number and proved that if degree 2|H|+1 of π-solvable irreducible complex linear group G with a π-Hall TI-subgroup H is not a prime power, then H is Abelian and normal in G.

scholarly journals On the uniform spread of almost simple linear groups

2013 ◽  
Vol 209 ◽  
pp. 35-109 ◽  
Author(s):  
Timothy C. Burness ◽  
Simon Guest
Keyword(s):  
Finite Group ◽  
Conjugacy Class ◽  
Simple Group ◽  
Nonnegative Integer ◽  
Finite Simple Group ◽  
Linear Groups ◽  
A Finite Group

AbstractLet G be a finite group, and let k be a nonnegative integer. We say that G has uniform spread k if there exists a fixed conjugacy class C in G with the property that for any k nontrivial elements x1,…,xk in G there exists y ∊ C such that G = ‹xi,y› for all i. Further, the exact uniform spread of G, denoted by u(G), is the largest k such that G has the uniform spread k property. By a theorem of Breuer, Guralnick, and Kantor, u(G) ≥ 2 for every finite simple group G. Here we consider the uniform spread of almost simple linear groups. Our main theorem states that if G = ‹PSLn (q),g› is almost simple, then u(G) ≥ 2 (unless G ≅ S6), and we determine precisely when u(G) tends to infinity as |G| tends to infinity.

scholarly journals Local Fusion Graphs and Sporadic Simple Groups

10.37236/4298 ◽  
2015 ◽  
Vol 22 (3) ◽  
Author(s):  
John Ballantyne ◽  
Peter Rowley
Keyword(s):  
Automorphism Group ◽  
Conjugacy Class ◽  
Simple Group ◽  
Simple Groups ◽  
Sporadic Simple Group ◽  
Odd Order ◽  
Vertex Set ◽  
Sporadic Simple Groups

For a group $G$ with $G$-conjugacy class of involutions $X$, the local fusion graph $\mathcal{F}(G,X)$ has $X$ as its vertex set, with distinct vertices $x$ and $y$ joined by an edge if, and only if, the product $xy$ has odd order. Here we show that, with only three possible exceptions, for all pairs $(G,X)$ with $G$ a sporadic simple group or the automorphism group of a sporadic simple group, $\mathcal{F}(G,X)$ has diameter $2$.

Sign in / Sign up

Export Citation Format

Share Document