Generating triples of involutions for lie-type groups over a finite field of odd characteristic. II

Abstract

We describe simple Lie-type groups of rank l ~_ 3 over a finite field of odd characteristic, generated by three involutions of which two are commuting. Previously, the answers to the similar questior#s were given for alternating groups, for Lie-type groups over a finite field of characteristic 2, and for Lie-type groups of rank l >_ 4 over a finite field of odd characteristic. These furnish a description of finite simple non-Abelian groups, distinct from 26 sporadic groups, generated by three involutions two of which are commuting, thus giving a partial answer to question 7.30

Chiral polyhedra and finite simple groups

“…In 1980, it was asked in the Kourovka Notebook (Problem 7.30) which finite simple groups have this property. This was solved by Nuzhin and others in : every non‐abelian finite simple group can be generated by three involutions, two of which commute, with the following exceptions: The groups and , although mentioned by Nuzhin as being generated by three involutions, two of which commute, have recently been discovered not to have such generating sets by M. Macaj and G. Jones (personal communication). Thus every finite simple group, apart from the above exceptions, is the automorphism group of an abstract regular polyhedron.…”