Explore advanced algorithms for finitely presented groups in this comprehensive lecture from the Hausdorff Trimester Program on Logic and Algorithms in Group Theory. Delve into fundamental computational methods for groups defined by finite presentations, including the Todd-Coxeter and Reidemeister-Schreier algorithms. Examine techniques for computing abelian quotients, identifying finite index subgroups, and generating subgroup presentations. Investigate the Dehn algorithm in small cancellation and hyperbolic groups, rewrite systems, and the Knuth-Bendix completion algorithm. Learn about finite state automata, automatic groups, and their applications in computing growth rates. Gain insights into approaches for solving conjugacy and generalized word problems in finitely presented groups. Enhance your understanding of group theory algorithms through practical examples and theoretical discussions.