Every four years, leading researchers gather to survey the latest developments in all aspects of group theory. Since 1981, the proceedings of those meetings have provided a regular snapshot of the state of the art in group theory and helped to shape the direction of research in the field. This volume contains selected papers from the 2013 meeting held in St Andrews. It begins with major articles from each of the four main speakers: Emmanuel Breuillard (Paris-Sud), Martin Liebeck (Imperial College London), Alan Reid (Texas) and Karen Vogtmann (Cornell). These are followed by, in alphabetical order, survey articles contributed by other conference participants, which cover a wide spectrum of modern group theory.

Table of Contents:
Introduction C. M. Campbell and E. F. Robertson; 1. Approximate subgroups and super-strong approximation Emmanuel Breuillard; 2. Width questions for finite simple groups Martin W. Liebeck; 3. Profinite properties of discrete groups Alan W. Reid; 4. GL(n,Z), Out(Fn) and everything in between: automorphism groups of RAAGs Karen Vogtmann; 5. Permutation groups and transformation semigroups: results and problems João Araújo and Peter J. Cameron; 6. New progress on factorized groups and subgroup permutability Milagros Arroyo-Jordá, Paz Arroyo-Jordá, Ana Martínez-Pastor and M. Dolores Pérez-Ramos; 7. A survey on the normalizer problem for integral group rings Andreas Bächle; 8. A survey on Clifford–Fischer Theory Ayoub B. M. Basheer and Jamshid Moori; 9. A generalisation on the solvability of finite groups with three class sizes for normal subgroups Antonio Beltrán and María José Felipe; 10. Automorphism groups of non-orientable Riemann surfaces E. Bujalance, F. J. Cirre, J. J. Etayo, G. Gromadzki and E. Martínez; 11. What are the C2-groups? Inna Capdeboscq and Christopher Parker; 12. Resurrecting Wells' exact sequence and Buckley's group action Jill Dietz; 13. Recent work on Beauville surfaces, structures and groups Ben Fairbairn; 14. Something for nothing: some consequences of the solution of the Tarski problems Benjamin Fine, Anthony Gaglione, Gerhard Rosenberger and Dennis Spellman; 15. The groups of projectivities in finite planes Theo Grundhöfer; 16. On the relation gap and relation lifting problem Jens Harlander; 17. Some results on products of finite subsets in groups Marcel Herzog, Patrizia Longobardi and Mercede Maj; 18. Formal languages and group theory Sam A. M. Jones and Richard M. Thomas; 19. On the Castelnuovo–Mumford regularity of the cohomology of fusion systems and of the Hochschild cohomology of block algebras Radha Kessar and Markus Linckelmann; 20. Recent advances on torsion subgroups of integral group rings Wolfgang Kimmerle and Alexander Konovalov; 21. On finite groups with small prime spectrum Anatoly S. Kondratiev and Igor V. Khramtsov; 22. Solvability criteria for finite loops and groups Emma Leppälä; 23. The rational subset membership problem for groups: a survey Markus Lohrey; 24. A survey of Milnor laws Olga Macedońska; 25. Capable p-groups Arturo Magidin and Robert Fitzgerald Morse; 26. On the normal structure of a finite group with restrictions on the maximal subgroups N. V. Maslova and D. O. Revin; 27. Certain monomial characters and their normal constituents Gabriel Navarro and Carolina Vallejo; 28. Recognition of finite quasi-simple groups by the degrees of their irreducible representations Hung Ngoc Nguyen and Hung P. Tong-Viet; 29. Generalized Baumslag–Solitar groups: a survey of recent progress Derek J. S. Robinson; 30. Zeta functions of groups and rings – recent developments Christopher Voll.