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

Abstract

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 N p -group if it contains a matrix g such that g p 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 N 2 -group exist for n > 1 . This paper is a major step toward the determination of all N p -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 N p -groups. This is done in the current paper, and the general case will be dealt with in a subsequent work.