Note on a conjecture of Coxeter

Abstract

Coxeter [1] has studied groups defined by the relationsand gives lists of finite groups known to be completely defined by such sets of relations. In a later paper [2] he shows that G3, n, p is finite if n, p are both even and satisfyand expresses the conjecture that the restriction to even values may be removed. The only case satisfying this inequality and not already known to be finite is G3, 7, 16. In this note we show that G3, 7, 16 is indeed finite, being of order 21504 = 210.3.7, by showing that its subgroupof index 2 is finite and of order 10752. Thus we add one entry to each of the lists of finite groups in Coxeter [1].