Derangements and eigenvalue-free elements in finite classical groups. (English) Zbl 0936.15020

Neumann, Peter M.; Praeger, Cheryl E.

J. Lond. Math. Soc., II. Ser. 58, No. 3, 564-586 (1998).

One of the questions addressed in this paper is: given the matrix ring \(M(n,q)\) of \(n\times n\) matrices over the field \(F_q\) with \(q\) elements, what is the probability \(\upsilon(M;n,q)\) that a randomly chosen matrix has no eigenvalues? Regarding different \(M\), the authors derive asymptotic expressions for \(\upsilon\), in the limit \(n\to\infty\), that has as the leading term \(e^{-1}\).

Reviewer: Alexei Khorunzhy (Khar’kov)

MSC:

15B52	Random matrices (algebraic aspects)
20G40	Linear algebraic groups over finite fields

Keywords:

derangements; eigenvalue-free elements; random matrices; finite classical groups; Euler’s generating functions; finite field; matrix ring