Construction of long root SL(2,q)-subgroups in black box groups

Sukru Yalcinkaya

paper

Construction of long root SL(2,q)-subgroups in black box groups

arXiv:1001.3184

Abstract

We present a one sided Monte--Carlo algorithm which constructs a long root $\sl_2(q)$-subgroup in $X / O_{p} (X)$ , where $X$ is a black-box group and $X / O_{p} (X)$ is a finite simple group of Lie type defined over a field of odd order $q = p^{k} > 3$ for some $k ⩾ 1$ . Our algorithm is based on the analysis of the structure of centralizers of involutions and can be viewed as a computational version of Aschbacher's Classical Involution Theorem. We also present an algorithm which determines whether the $p$ -core (or "unipotent radical") $O_{p} (X)$ of a black-box group $X$ is trivial or not, where $X / O_{p} (X)$ is a finite simple classical group of odd characteristic $p$ . This answers a well-known question of Babai and Shalev.

37 pages, submitted to Journal of Algebra