Abelian coverings of finite general linear groups and an application to their non-commuting graphs.

Azad, Azizollah ; Iranmanesh, Mohammad A. ; Praeger, Cheryl E. ; Spiga, Pablo

Journal of Algebraic Combinatorics, Tome 34 (2011) no. 4, pp. 683-710.

Résumé

Summary: In this paper we introduce and study a family $A _{ n}( q)$ mathcalA_n$(q)$ of abelian subgroups of GL $_{ n}( q) {\rm $GL_n$(q)$ covering every element of GL $_{ n}( q) {\rm $GL_n$(q)$. We show that $A _{ n}( q)$ mathcalA_n$(q)$ contains all the centralizers of cyclic matrices and equality holds if $q> n$. For $q>2$, we obtain an infinite product expression for a probabilistic generating function for | $A _{ n}( q)$ | |mathcalA_n$(q)$|. This leads to upper and lower bounds which show in particular that $c _{1} q ^{ - n} \sterling \frac | A _{ n}( q)$ | | GL $_{ n}( q) | \sterling c _{2} q ^{ - n}$ c_1q^-n$\leq $frac|mathcalA_$n(q)$||mathrmGL_$n(q)|}\leq c$_2q^-n for explicit positive constants $c _{1}, c _{2}$. We also prove that similar upper and lower bounds hold for $q=2$. A subset $X$ of a finite group $G$ is said to be pairwise non-commuting if $xy \textonesuperior yx$ xynot=yx for distinct elements $x, y$ in $X$. As an application of our results on $A _{ n}( q)$ mathcalA_n$(q)$, we prove lower and upper bounds for the maximum size of a pairwise non-commuting subset of GL $_{ n }( q)$. (This is the clique number of the non-commuting graph.) Moreover, in the case where $q> n$, we give an explicit formula for the maximum size of a pairwise non-commuting set.

Détail

Keywords: keywords general linear group, cyclic matrix, non-commuting subsets of finite groups, non-commuting graph