Constructive recognition of 𝑃𝑆𝐿(2,𝑞)

Abstract

Existing black box and other algorithms for explicitly recognising groups of Lie type over G F ( q ) \mathrm {GF}(q) have asymptotic running times which are polynomial in q q , whereas the input size involves only log ⁡ q \log q . This has represented a serious obstruction to the efficient recognition of such groups. Recently, Brooksbank and Kantor devised new explicit recognition algorithms for classical groups; these run in time that is polynomial in the size of the input, given an oracle that recognises P S L ( 2 , q ) \mathrm {PSL}(2,q) explicitly. The present paper, in conjunction with an earlier paper by the first two authors, provides such an oracle. The earlier paper produced an algorithm for explicitly recognising S L ( 2 , q ) \mathrm {SL}(2,q) in its natural representation in polynomial time, given a discrete logarithm oracle for G F ( q ) \mathrm {GF}(q) . The algorithm presented here takes as input a generating set for a subgroup G G of G L ( d , F ) \mathrm {GL}(d,F) that is isomorphic modulo scalars to P S L ( 2 , q ) \mathrm {PSL}(2,q) , where F F is a finite field of the same characteristic as G F ( q ) \mathrm {GF}(q) ; it returns the natural representation of G G modulo scalars. Since a faithful projective representation of P S L ( 2 , q ) \mathrm {PSL}(2,q) in cross characteristic, or a faithful permutation representation of this group, is necessarily of size that is polynomial in q q rather than in log ⁡ q \log q , elementary algorithms will recognise P S L ( 2 , q ) \mathrm {PSL} (2,q) explicitly in polynomial time in these cases. Given a discrete logarithm oracle for G F ( q ) \mathrm {GF}(q) , our algorithm thus provides the required polynomial time oracle for recognising P S L ( 2 , q ) \mathrm {PSL}(2,q) explicitly in the remaining case, namely for representations in the natural characteristic. This leads to a partial solution of a question posed by Babai and Shalev: if G G is a matrix group in characteristic p p , determine in polynomial time whether or not O p ( G ) O_p(G) is trivial.