Writing Elements of PSL(2, q ) as Commutators
Darryl McCullough, Marcus Wanderley · Communications in Algebra · 2011
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).