Writing Elements of PSL(2, q) as Commutators

Abstract

We prove that for q ≥ 13, an element A of SL(2, q) is the commutator of a generating pair if and only if A ≠ −I and the trace of A is not 2. Consequently, when q is odd and q ≥ 13, every nontrivial element of PSL(2, q) is the commutator of a generating pair, and when q is even, an element of PSL(2, q) is the commutator of a generating pair if and only if its trace is not 0. The proof of these results also leads to an improved lower bound on the number of T-systems of generating pairs of PSL(2, q).

Key Words:

• Commutator
• Projective
• Special linear group
• Trace
• T-system

2000 Mathematics Subject Classification:

• Primary 20G40

ACKNOWLEDGMENTS

The authors gratefully acknowledge the support of many sources in the course of this work. Travel of the first author was supported by National Science Foundation grant DMS-0102463, by the University of Oklahoma College of Arts and Sciences, and by the University of Oklahoma Office of Research Administration. The work of both authors was supported by the Universidade Federal de Pernambuco Summer Program in Recife. We are grateful to Martin Evans for providing us a copy of [5], and to the referee for providing a number of improvements to the original manuscript. We owe special thanks to Gareth Jones, who showed us the proof of the formula for Ψ _q that we gave above.

The first author was supported in part by NSF grants DMS-0102463 and DMS-0802424.

Notes

Communicated by M. Dixon.