Primitive normal matrices and covering numbers of finite groups

A primitive matrix is a square matrix M with nonnegative real entries such that the entries of M-s are all positive for some positive integer s. The smallest such s is called the primitivity index of M. Primitive matrices of normal type (namely: MMT and (MM)-M-T have the same zero entries) occur naturally in studying the so called "conjugacy-class covering number" and "character covering number" of a finite group. We show that if M is a primitive n x n matrix of normal type with minimal polynomial of degree in, then the primitivity index of M is at most ([n/2] + 1)(m - 1). This bound is then applied to improve known bounds for the various covering numbers of finite groups. (c) 2005 Elsevier Inc. All rights reserved.