Efficient Presentations For PSL(2,p) and Related Groups

The deficiency \(n\) of a group \(G\) is the largest integer such that \(G\) has a presentation with \(m\) generators and \(m-n\) relations. If \(G\) is finite, then \(n\) is a negative number.

An efficient presentation of a finite group \(G\) has \(m-r\) relations, where \(r\) is the rank of the Schur multiplier, which is defined thus:

\[ M(G) = \frac{(F' \cap R)}{[F,R]} \]

where \(F\) is the free group on the set of generators of \(G\), and \(R\) is the normal closure of the relations holding in \(G\). The groups \(\mathrm{PSL}(2,p)\) have Schur multiplier \(\mathbb{Z}_2\), which is cyclic, and hence of rank 1. The groups \(\mathrm{SL}(2,p)\) has trivial Schur multiplier.

Thus an efficient presentation of \(\mathrm{PSL}(2,p)\) will have 2 generators and 3 relators, whereas \(\mathrm{SL}(2,p)\) will have 2 generators and 3 relators.

The projective special linear groups \(\mathrm{PSL}(2,p)\) are obviously defined by

\[ \mathrm{PSL}(2,p) = \mathrm{SL}(2,p) / \{ \pm I \} \]

where