On the Computation of Conjugacy Classes of Chevalley Groups

1996 ◽  
Vol 7 (3) ◽  
pp. 221-234
Author(s):  
Peter Fleischmann ◽  
Ingo Janiszczak
Keyword(s):  
Conjugacy Classes ◽  
Chevalley Groups

scholarly journals Calculating conjugacy classes in Sylow -subgroups of finite Chevalley groups of rank six and seven

2014 ◽  
Vol 17 (1) ◽  
pp. 109-122 ◽  
Author(s):  
Simon M. Goodwin ◽  
Peter Mosch ◽  
Gerhard Röhrle
Keyword(s):  
Computer Program ◽  
Explicit Formula ◽  
Chevalley Group ◽  
Conjugacy Classes ◽  
Chevalley Groups ◽  
Sylow Subgroups ◽  
Finite Chevalley Group ◽  
Integer Coefficients ◽  
The Third

AbstractLet$G(q)$be a finite Chevalley group, where$q$is a power of a good prime$p$, and let$U(q)$be a Sylow$p$-subgroup of$G(q)$. Then a generalized version of a conjecture of Higman asserts that the number$k(U(q))$of conjugacy classes in$U(q)$is given by a polynomial in$q$with integer coefficients. In [S. M. Goodwin and G. Röhrle,J. Algebra321 (2009) 3321–3334], the first and the third authors of the present paper developed an algorithm to calculate the values of$k(U(q))$. By implementing it into a computer program using$\mathsf{GAP}$, they were able to calculate$k(U(q))$for$G$of rank at most five, thereby proving that for these cases$k(U(q))$is given by a polynomial in$q$. In this paper we present some refinements and improvements of the algorithm that allow us to calculate the values of$k(U(q))$for finite Chevalley groups of rank six and seven, except$E_7$. We observe that$k(U(q))$is a polynomial, so that the generalized Higman conjecture holds for these groups. Moreover, if we write$k(U(q))$as a polynomial in$q-1$, then the coefficients are non-negative.Under the assumption that$k(U(q))$is a polynomial in$q-1$, we also give an explicit formula for the coefficients of$k(U(q))$of degrees zero, one and two.

scholarly journals Twisted Conjugacy Classes in Chevalley Groups

2015 ◽  
Vol 53 (6) ◽  
pp. 481-501 ◽  
Author(s):  
T. R. Nasybullov
Keyword(s):  
Conjugacy Classes ◽  
Chevalley Groups ◽  
Twisted Conjugacy

scholarly journals Twisted conjugacy classes in twisted Chevalley groups

2020 ◽  
pp. 2250052
Author(s):  
Sushil Bhunia ◽  
Pinka Dey ◽  
Amit Roy
Keyword(s):  
Characteristic Zero ◽  
Conjugacy Classes ◽  
Transcendence Degree ◽  
Chevalley Groups ◽  
Twisted Conjugacy

Let [Formula: see text] be a group and [Formula: see text] be an automorphism of [Formula: see text]. Two elements [Formula: see text] are said to be [Formula: see text]-twisted conjugate if [Formula: see text] for some [Formula: see text]. We say that a group [Formula: see text] has the [Formula: see text]-property if the number of [Formula: see text]-twisted conjugacy classes is infinite for every automorphism [Formula: see text] of [Formula: see text]. In this paper, we prove that twisted Chevalley groups over a field [Formula: see text] of characteristic zero have the [Formula: see text]-property as well as the [Formula: see text]-property if [Formula: see text] has finite transcendence degree over [Formula: see text] or [Formula: see text] is periodic.

scholarly journals Involutions in Chevalley groups over fields of even order

1976 ◽  
Vol 63 ◽  
pp. 1-91 ◽  
Author(s):  
Michael Aschbacher ◽  
Gary M. Seitz
Keyword(s):  
Chevalley Group ◽  
Conjugacy Classes ◽  
Chevalley Groups ◽  
Even Order ◽  
Characteristic 2

Let G = G(q) be a Chevalley group defined over a field Fq of characteristic 2. In this paper we determine the conjugacy classes of involutions in Aut(G) and the centralizers of these involutions. This study was begun in the context of a different problem.

scholarly journals Calculating conjugacy classes in Sylow p-subgroups of finite Chevalley groups

2009 ◽  
Vol 321 (11) ◽  
pp. 3321-3334 ◽  
Author(s):  
Simon M. Goodwin ◽  
Gerhard Röhrle
Keyword(s):  
Conjugacy Classes ◽  
Chevalley Groups
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