Black box exceptional groups of Lie type. (Q2847127)

This paper takes a further step in the development of computational algorithms for the recognition and investigation of finite simple groups and linear groups definable over finite fields. In the case considered in this paper, the given group \(G\) is assumed to be a perfect central extension of an exceptional simple group \(H\) of Lie type, of (twisted) rank greater than \(1\) over a finite field of known order \(q\), other than \(^2F_4(q)\). Also the group is a `black-box' group, not given in any particular representation.NEWLINENEWLINE The authors describe a Las Vegas algorithm that recognises the group \(G/Z(G)\), and gives a constructive isomorphism between it and the identified exceptional simple group \(H\), as well as a corresponding epimorphism from the universal cover \(\widehat H\) (of \(H\)) onto \(G\).NEWLINENEWLINE The approach involves finding a long root element and then building a subgroup isomorphic to \(\mathrm{SL}(3,q)\), and also a subgroup \(\mathrm{Spin}^-_8(q)\) when the Lie rank is greater than 2. Pieces of these groups are used to obtain the centraliser of a subgroup isomorphic to \(\mathrm{SL}(2,q)\) generated by long root groups. Then (in contrast to approach for classical groups) the algorithm finds all of the root groups corresponding to a root system, and verifies the standard commutator relations that define these groups. The corresponding presentation guarantees the Las Vegas nature of the algorithm.NEWLINENEWLINE The algorithm does not run in polynomial time in the input, as the complexity (resulting from two of the main steps) involves \(q\), but the authors propose some possible variants which might lead to more effective implementations.