Some Two-Dimensional Unitary Groups Generated by Three Reflections

Abstract

Shephard and Todd (5) give generators for the finite primitive irreducible groups generated by two unitary reflections in U2. It is the purpose of the present paper to give generating reflections, and defining relations in terms of these reflections, for the seven such groups requiring three generating reflections, that is, for their nos. 7, 11, 12, 13, 15, 19, 22. The reflections are chosen whenever possible so that their product has the property suggested by Theorem 5.4 of (5). That is, except for no. 15, the period of the product of the three generating reflections is h = m2 + 1, and the characteristic roots of this product are 2πim1/h and 2πim2/h, where m1 and m2 are the “exponents“ (5, p. 282) of the group. The reason for the impossibility of such a choice for no. 15 is given in § 4. In § 5 the homomorphisms between these groups and certain groups of motions in elliptic 3-space are determined.