'Groups St Andrews 2005' was held in the University of St Andrews in August 2005 and this first volume of a two-volume book contains selected papers from the international conference. Four main lecture courses were given at the conference, and articles based on their lectures form a substantial part of the Proceedings. This volume contains the contributions by Peter Cameron (Queen Mary, London) and Rostislav Grogorchuk (Texas A&M, USA). Apart from the main speakers, refereed survey and research articles were contributed by other conference participants. Arranged in alphabetical order, these articles cover a wide spectrum of modern group theory. The regular Proceedings of Groups St Andrews conferences have provided snapshots of the state of research in group theory throughout the past 25 years. Earlier volumes have had a major impact on the development of group theory and it is anticipated that this volume will be equally important.

Introduction; 1. Aspects of infinite permutation groups Peter J. Cameron; 2. Self-similarity and branching in group theory Rostislav Grigorchuk and Zoran Sunic; 3. On surface groups: motivating examples in combinatorial group theory Peter Ackermann, Benjamin Fine and Gerhard Rosenberger; 4. Nilpotent p-algebras and factorized p-groups Bernhard Amberg and Lev Kazarin; 5. Classification of finite groups by the number of element centralizers Ali Reza Ashrafi and Bijan Taeri; 6. Algorithmic use of the Mal'cev correspondence Bjoern Assmann; 7. Minimal but inefficient presentations for semi-direct products of finite cyclic monoids Firat Ates and A. Sinan Ç evik; 8. The modular isomorphism problem for finite p-groups with a cyclic subgroup of index p2 Czeslaw Baginski and Alexander Konovalov; 9. On one-generated formations A. Ballester-Bolinches, Clara Calvo and R. Esteban-Romero; 10. New results on products of finite groups A. Ballester-Bolinches, John Cossey and M. C. Pedraza-Aguilera; 11. Radical locally finite T-groups A. Ballester-Bolinches, H. Heineken and Tatiana Pedraza; 12. Explicit tilting complexes for the Broué conjecture on 3-blocks Ayala Bar-Ilan, Tzviya Berrebi, Genadi Chereshnya, Ruth Leabovich, Mikhal Cohen and Mary Schaps; 13. Conjugacy classes of p-regular elements in p-solvable groups Antonio Beltrán and Maria José Felipé; 14. An algorithm for the unit group of the Burnside ring of a finite group Robert Boltje and Götz Pfeiffer; 15. Integral group ring of the first Mathieu simple group Victor Bovdi and Alexander Konovalov; 16. Embedding properties in direct products B. Brewster, A. Martinez-Pastor and M. D. Pérez-Ramos; 17. Malcev presentations for subsemigroups of groups - a survey Alan J. Cain; 18. Finite groups with extremal conditions on sizes of conjugacy classes and on degrees of irreducible characters David Chillag and Marcel Herzog; 19. Conjugacy class structure in simple algebraic groups Martin Cook; 20. On automorphisms of products of groups Jill Dietz; 21. Linear groups with infinite central dimension Martyn R. Dixon and Leonid A. Kurdachenko; 22. G-automata, counter languages and the Chomsky hierarchy Murray Elder; 23. An embedding theorem for groups universally equivalent to free nilpotent groups Benjamin Fine, Anthony M. Gaglione and Dennis Spellman; 24. Irreducible word problems in groups Ana R. Fonseca, Duncan W. Parkes and Richard M. Thomas; 25. Recent growth results Eric M. Freden and Teresa Knudson.