In this paper we introduce and study a family An(q) of abelian subgroups of GLn(q) covering every element of GLn(q). We show that An(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 |An(q)|. This leads to upper and lower bounds which show in particular that c1q-n ≤ |A n(q)| |GLn(q)| ≤ c2q-n for explicit positive constants c1, c2. 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 yx for distinct elements x,y in X. As an application of our results on A n(q), we prove lower and upper bounds for the maximum size of a pairwise non-commuting subset of GLn(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. © 2011 Springer Science+Business Media, LLC.

Azad, A., Iranmanesh, M., Praeger, C., Spiga, P. (2011). Abelian coverings of finite general linear groups and an application to their non-commuting graphs. JOURNAL OF ALGEBRAIC COMBINATORICS, 34(4), 683-710 [10.1007/s10801-011-0288-2].

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

SPIGA, PABLO
Ultimo
2011

Abstract

In this paper we introduce and study a family An(q) of abelian subgroups of GLn(q) covering every element of GLn(q). We show that An(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 |An(q)|. This leads to upper and lower bounds which show in particular that c1q-n ≤ |A n(q)| |GLn(q)| ≤ c2q-n for explicit positive constants c1, c2. 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 yx for distinct elements x,y in X. As an application of our results on A n(q), we prove lower and upper bounds for the maximum size of a pairwise non-commuting subset of GLn(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. © 2011 Springer Science+Business Media, LLC.
Articolo in rivista - Articolo scientifico
Cyclic matrix; General linear group; Non-commuting subsets of finite groups-Non-commuting graph; Discrete Mathematics and Combinatorics; Algebra and Number Theory
English
683
710
28
Azad, A., Iranmanesh, M., Praeger, C., Spiga, P. (2011). Abelian coverings of finite general linear groups and an application to their non-commuting graphs. JOURNAL OF ALGEBRAIC COMBINATORICS, 34(4), 683-710 [10.1007/s10801-011-0288-2].
Azad, A; Iranmanesh, M; Praeger, C; Spiga, P
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10281/133186
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