Black Box Recognition of the Suzuki groups

Abstract

We present a black box algorithm that constructs standard generators for the Suzuki groups $Sz(q)$, where $q = 2^{2m+1}$ for some $m > 0$. The algorithm is one-sided Monte Carlo, with no false positives. We also present a black box algorithm that performs constructive membership testing in $Sz(q)$, and writes an element as a straight line program in the standard generators. Finally, we give a presentation for $Sz(q)$ that is efficient to verify. The algorithms have been implemented in the computer algebra system Magma.