Primitive normal matrices and covering numbers of nite groups

A primitive matrix is a square matrix M with nonnegative real entries such that the entries of M 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: MM and MM have the same zero entries) occur naturally in studying the so called ”conjugacy-class covering number”and ”character covering number”of a …nite 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 n 2 + 1 (m 1). This bound is then applied to improve known bounds for the various covering numbers of …nite groups.