Some definitions of Klein's simple group of order 168 and other groups

Abstract

In a previous note [3], Mennicke and I showed that the relations(8, 7|2, 3): A8=B7=(AB)2=(AB)2=(A-1B)2=(A-1B)3=Edefine a group of order 10752. As we remarked, the results of §§ 2, 3 of that note are not restricted in their application to this group; they apply to the group[3, 7]+: B7=(AB)2=(A-1B)3=Eand to any factor group of this group which in turn has Klein's simple group of order 168, defined by(4,7|2, 3): A4=B7=(AB)2=(A-1B)3=E,as a factor group. In this note I use these results to establish alternative “weaker” definitions for Klein's group and for two groups discussed by Sinkov [4], namely (8, 7|2, 3) defined above and a factor group of this group of order 1344. These latter groups are eloquently discussed by Coxeter [1].