A property of triangle groups

The F .groups a re the g roups possessing fai t hful rep rese nt at io ns by Fuchsian groups of the firs t kind ; the ir presenta tions a re known eXI)licit ly. Amo ng the F ·groups a re the well · know n triangle groups C = {x, ylx"=Y"= {xy}'·= l}. H p , q, r are dis tin ct pri me intege rs, eve ry prope r norm a l su bgroup of fin it e inde x in C has no e le me nts of fini te order. In t hi s pape r it is proved that a mong the F ·groups onl y the triangle groups with di s tinct prime p , q, r have thi s prope rt y.