On Groups Generated by Elements of Pirme Order

Abstract

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 pth power of the determinant equal to 1 over a field F containing a primitive pth 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 ℝ and ℂ). For a universal p-Coxeter group of rank ≥ 2 for p ≥ 3 or of rank ≥ 3 for p = 2 there is no bound k.