Minimum polynomials of the elements of prime order in representations of quasi-simple groups

We determine the irreducible representations of quasi-simple groups in which some element of prime order p has less than p distinct eigenvalues. Let p be a prime greater than 2. Let C denote the field of complex numbers, GL(n,C) the group of all (n×n)-matrices over C. Let G⊆GL(n,C) be a finite irreducible subgroup, Z(G) the center of G. Let p>2 be a prime. We call G an Np-group if it contains a matrix g such that gp is scalar, g has at most p−1 distinct eigenvalues and g does not belong to a proper normal subgroup of G. We assume p>2 as no N2-group exist for n>1. This paper is a major step toward the determination of all Np-groups. This will serve for recognition of finite linear groups containing a given matrix with the above property for some p. The bulk of the work is to determine quasi-simple Np-groups. This is done in the current paper, and the general case will be dealt with in a subsequent work.