ON GROUPS GENERATED BY ELEMENTS OF PRIME ORDER L. Grunenfelder, T. Koˇ sir, M. Omladiˇ c, and H. Radjavi In the first part of the paper we give a characterization of groups generated by elements of fixed prime order p. In the second part we study the group G (p) n of n × n matrices with the p-th power of the determinant equal to 1 over a field F containing a primitive p-th root of 1. It is known that the group G (2) n of n × n matrices of determinant + - 1 over a field F and the group SL n (F ) are generated by their involutions and that each element in these groups is a product of four involutions. We consider some subgroups G of G (p) n and study the following problems: Is G generated by its elements of order p ? If so, is every element of G a product of k elements of order p for some fixed integer k ? We show that G (p) n and SL n (F ) are generated by their elements of order p and that the bound k exists and is equal to 4. We show that every universal p-Coxeter group has faithful two-dimensional representations over many fields F (including and ). For a universal p-Coxeter group of rank ≥ 2 for p ≥ 3 or of rank ≥ 3 for p = 2 there is no bound k. : 15A23, 20F55, 51N30 : groups, elements of prime order, matrix groups, factorization, special linear group, universal Coxeter group 1. Introduction The purpose of this paper is twofold. We first give a characterization of general groups generated by elements of fixed prime order p and discuss the consequences for simple, solvable and nilpotent groups. In section 3 we study universal Coxeter groups [H, §5.1] and their generalization to universal p-Coxeter groups, i.e. groups G generated by a set X subject only to relations x p = 1 for all x ∈ X . We show that every universal p-Coxeter group G, of finite or infinite rank r, has a two-dimensional faithful representation over many fields (including R and C). Note that the standard geometric representation of Coxeter groups is on an r-dimensional vector space [H, §5.4]. Our two-dimensional faithful representation of G for r ≥ 2 is of minimal dimension since G is not commutative. In the rest of the paper we concentrate on matrix groups generated by elements of order p. Let G (p) n be the subgroup of the general linear group GL n (F ) consisting of all matrices A with (det A) p = 1. The case of matrix groups G (2) n generated by involutions, i.e. matrices J with J 2 = I , has been studied previously. In [GHR] the authors show that G (2) n is generated by its involutions; moreover, every element in G (2) n is a product of four involutions Research supported in part by the NSERC of Canada and by the Ministry of Science and Technology of Slovenia. Typeset by A M S-T E X 1