This volume contains the proceedings of the NATO Advanced Study Institute on Finite and Locally Finite Groups held in Istanbul, Turkey, 14-27 August 1994, at which there were about 90 participants from some 16 different countries. The ASI received generous financial support from the Scientific Affairs Division of NATO. INTRODUCTION A locally finite group is a group in which every finite set of elements is contained in a finite subgroup. The study of locally finite groups began with Schur's result that a periodic linear group is, in fact, locally finite. The simple locally finite groups are of particular interest. In view of the classification of the finite simple groups and advances in representation theory, it is natural to pursue classification theorems for simple locally finite groups. This was one of the central themes of the Istanbul conference and significant progress is reported herein. The theory of simple locally finite groups intersects many areas of group theory and representation theory, so this served as a focus for several articles in the volume. Every simple locally finite group has what is known as a Kegel cover. This is a collection of pairs {(G , Ni) liE I}, where I is an index set, each group Gi is finite, i Ni

Locally Finite Groups

Part 1 (ISBN 978-3-7568-0801-4) of the Trilogy is based on the BoD-Book "Characterising locally finite groups satisfying the strong Sylow Theorem for the prime p - Revised edition" (see ISBN 978-3-7562-3416-5). The First edition of Part 1 (see ISBN 978-3-7543-6087-3) removes the highlights in light green of the Revised edition, adds 14 pages to the AGTA paper and 10 pages to the Revised edition. It includes Reference [11] resp. [10] as Appendix 1 resp. Appendix 2 and calls to mind Professor Otto H. Kegel's contribution to the conference Ischia Group Theory 2016. The Second edition introduces a uniform page numbering, adds page numbers to the appendices, improves 19 pages, adds Pages 109 to 112 and a Table of Contents. Part 2 (ISBN 978-3-7543-3642-8) of the Trilogy is based on the author's research paper "About the Strong Sylow Theorem for the Prime p in Simple Locally Finite Groups". We first give an overview of simple locally finite groups and reduce their Sylow theory for the prime p to a conjecture of Prof. Otto H. Kegel about the rank-unbounded ones of the 19 known families of finite simple groups. Part 2 introduces a new scheme to describe the 19 families, the family T of types, defines the rank of each type, and emphasises the rôle of Kegel covers. This part presents a unified picture of known results and is the reason why our title starts with "About". We then apply new ideas to prove the conjecture for the alternating groups (see Page ii). Thereupon we remember Kegel covers and *-sequences. Finally we suggest a plan how to prove the conjecture step-by-step which leads to further conjectures thereby unifying Sylow theory in locally finite simple groups with Sylow theory in locally finite and p-soluble groups. In Part 3 (ISBN 978-3-7578-6001-1) of the Trilogy we continue the program begun in [10] to optimise along the way 1) its Theorem about the first type "An" of infinite families of finite simple groups step-by-step to further types by proving it for the second type "A = PSLn". We start with proving the Conjecture 2 of [10] about the General Linear Groups by using new ideas (see Page ii), and then break down this insight to the Special Linear and the PSL Groups. We close with suggestions for future research regarding the remaining rank-unbounded types (the "Classical Groups") and the way 2), the (locally) finite and p-soluble groups, and Augustin-Louis Cauchy's and Évariste Galois' contributions to Sylow theory in finite groups.

Part 1 of the Trilogy "Characterising Locally Finite Groups Satisfying the Strong Sylow Theorem for the Prime p" & "About the Strong Sylow Theorem for the Prime p in Simple Locally Finite Groups" & "The Strong Sylow Theorem for the Prime p in Projective Special Linear Locally Finite Groups" is based on the beauteous BoD-Book "Characterising locally finite groups satisfying the strong Sylow Theorem for the prime p - Revised edition" (see ISBN 978-3-7562-3416-5) which in turn has been based on the author's research paper "Characterising Locally Finite Groups Satisfying the Strong Sylow Theorem for the Prime p" that was published on pp. 13-39 of Volume 13 of the gratifyingly open access mathematical journal Advances in Group Theory and Applications (AGTA) (see https://www.advgrouptheory.com/ journal/#read). Part 1 removes the highlights in light green of the Revised edition and adds the albeit considerably improved Pages i to vi, Pages 26a to 26f, and Pages xiii to xviii to the AGTA paper. In addition it adds the ten new Pages xv to xxiv to the Revised edition and thus renumbers the Pages xv to xviii into the Pages xxv to xxviii. It includes Reference [11] as Appendix 1 and Reference [10] as Appendix 2. Finally it calls to mind Prof. Otto H. Kegel's fine contribution to the conference Ischia Group Theory 2016.

This book is concerned with the generalizations of Sylow theorems and the related topics of formations and the fitting of classes to locally finite groups. It also contains details of Sunkov's and Belyaev'ss results on locally finite groups with min-p for all primes p. This is the first time many of these topics have appeared in book form. The body of work here is fairly complete.

The Revised edition is based on the author's paper "Characterising Locally Finite Groups Satisfying the Strong Sylow Theorem for the Prime p" which has been published on pp. 13-39 of Volume 13 of the very fine open access mathematical journal Advances in Group Theory and Applications (AGTA) (see https://www.advgrouptheory.com/journal/#read). For that paper the author has transferred the copyright to AGTA. The Revised edition introduces quite a number of corrections and embellishments, highlighted in light green, which could not have been considered by AGTA, and especially a much more beautiful line and page formatting. For these enhancements the author has kept the copyright. The Revised edition adds Pages i to vi, Pages 26a to 26f and Pages xiii to xviii to the AGTA paper which either are required for a book - the front matter (die "Titelei") - or describe related aspects and background which cannot be published in a mathematical journal. The Revised edition incorporates major revisions by the author and by editors as well as some supplementary material designed to bring the research paper up to date.

Part 1 of the Trilogy "Characterising Locally Finite Groups Satisfying the Strong Sylow Theorem for the Prime p" & "About the Strong Sylow Theorem for the Prime p in Simple Locally Finite Groups" & "The Strong Sylow Theorem for the Prime p in Projective Special Linear Locally Finite Groups" is based on the beauteous BoD-Book "Characterising locally finite groups satisfying the strong Sylow Theorem for the prime p - Revised edition" (see ISBN 978-3-7562-3416-5) which in turn has been based on the author's research paper "Characterising Locally Finite Groups Satisfying the Strong Sylow Theorem for the Prime p" that was published on pp. 13-39 of Volume 13 of the open access mathematical journal Advances in Group Theory and Applications (AGTA) (look at https://www.advgrouptheory.com/journal/#read). The First edition of Part 1 (see ISBN 978-3-7543-6087-3) removes the highlights in light green of the Revised edition and adds the albeit fairly considerably improved Pages i to vi and Pages 27 to 34 to the AGTA paper. In addition Part 1 adds the ten new Pages 35 to 44 to the Revised edition and therefore has to renumber the Pages xv to xviii into the Pages 45 to 48. It includes the Reference [11] as Appendix 1 and the Reference [10] as Appendix 2. Finally it calls to mind Professor Otto H. Kegel's fine contribution to the conference Ischia Group Theory 2016. The Second edition introduces a uniform page numbering, adds page numbers to the appendices, improves Pages iv and v, Page 22, Pages 26 to 34 and Pages 39, 45, 49, 50, 75, 76, 105 and 106, adds Pages 109 to 112, and adds a two-page Table of Contents of the Trilogy. For a review of the trilogy see [16].