The L₂-quotient algorithm by Plesken and Fabianska finds for a given finitely presented group G all quotients of G which are isomorphic to some finite simple group PSL(2,q), simultaneously for every prime power q. The L₃-U₃-quotient algorithm does the same for the quotients PSL(3,q) and PSU(3,q).

After giving a motivation for these algorithms, I will describe the basic ideas which combine methods from representation theory and commutative algebra. At the end, I will try to give a live demonstration of the algorithms with several examples.