A question by Graham Higman concerning quotients of the (2, 3, 7) triangle group

Abstract

Using coset diagrams it is shown that for every sufficiently large positive integer n, both the alternating group A n and the symmetric group S n are quotients of the group G 3,7, 168 = 〈A, B, C ¦ A 3 = B 7 = C 168 = (AB) 2 = (BC) 2 = (CA) 2 = (ABC) 2 = 1〉 , and hence that all but finitely many of the alternating groups A n can be generated by elements x and y satisfying the relations x 2 = y 3 = ( xy) 7 = ( x −1 y −1 xy) m = 1, where m = 84. This is the smallest known value of m for which this phenomenon occurs.