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 Ms 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 MTM 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 × n matrix of normal type with minimal polynomial of degree m, then the primitivity index of M is at most n2+1(m-1). This bound is then applied to improve known bounds for the various covering numbers of finite groups.