Two-Generator Two-Relation Presentations for Special Linear Groups

A finite group defined by n generators and m relations must have m ⩾ n. A finite group is said to have deficiency zero if it has a presentation with n generators and n relations. In 1907 Schur [13] proved important results showing that certain finite groups could not have deficiency zero presentations. Let SL(2, p) denote the group of 2 X 2 matrices of determinant 1 over the field GF(p) p an odd prime, and put PSL(2, p) = SL(2, p)/{±I}. Now PSL(2, p) and SL(2,p) can be generated by two elements, but Schur’s result showed that PSL(2, p) required at least three relations. However, the possibility of a 2-generator 2-relation presentation for SL(2, p) was not excluded.