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Part 2 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 author's research paper "About the Strong Sylow Theorem for the Prime p in Simple Locally Finite Groups". This very beautiful and pioneering manuscript had been submitted for peer reviewing to the open access journals Advances in Group Theory and Applications (AGTA) (see https://www.advgrouptheory.com/journal/) and Science Research Association (SCIREA) Journal of Mathematics (see https://www.scirea.org/journal/Mathematics) but was very regrettably rejected by both of them (with ridiculous arguments). We first give a profound overview of the structure of simple groups and in particular of the simple locally finite groups and reduce their Sylow theory for the prime p to a famous conjecture of Prof. Otto H. Kegel (see [16], Theorem 2.4: "Let the p-subgroup P be a p-uniqueness subgroup in the finite simple group S which belongs to one of the seven rank-unbounded families. Then the rank of S is bounded in terms of P.") 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 all proofs of which are by reference and it is the actual reason why our title starts with "About". We then apply beautiful new ideas to prove the conjecture for the alternating groups (see Page ii). Thereupon we are remembering Kegel covers and *-sequences. Finally we suggest a plan how to prove and even how to optimise the conjecture step-by-step or peu à peu which leads to further quite tough conjectures thereby unifying Sylow theory in locally finite simple groups with Sylow theory in locally finite and p-soluble groups. For any unexplained terminology we refer to [6].
Part 2 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 author's research paper "About the Strong Sylow Theorem for the Prime p in Simple Locally Finite Groups". This very beautiful and pioneering manuscript had been submitted for peer reviewing to the open access journals Advances in Group Theory and Applications (AGTA) (see https://www.advgrouptheory.com/ journal/) and Science Research Association (SCIREA) Journal of Mathematics (see https://www.scirea.org/ journal/Mathematics) but was very regrettably rejected by both of them (with ridiculous arguments). We first give a profound overview of the structure of simple groups and in particular of the simple locally finite groups and reduce their Sylow theory for the prime p to a famous conjecture of Prof. Otto H. Kegel (see [16], Theorem 2.4: "Let the p-subgroup P be a p-uniqueness subgroup in the finite simple group S which belongs to one of the seven rank-unbounded families. Then the rank of S is bounded in terms of P.") 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 all proofs of which are by reference and it is the actual reason why our title starts with "About". We then apply beautiful new ideas to prove the conjecture for the alternating groups (see Page ii). Thereupon we are remembering Kegel covers and *-sequences. Finally we suggest a plan how to prove and even how to optimise the conjecture step-by-step or peu à peu which leads to further quite tough conjectures thereby unifying Sylow theory in locally finite simple groups with Sylow theory in locally finite and p-soluble groups. For any unexplained terminology we refer to [6].
This research paper continues [15]. We begin with giving a profound overview of the structure of arbitrary simple groups and in particular of the simple locally finite groups and reduce their Sylow theory for the prime p to a quite famous conjecture by Prof. Otto H. Kegel (see [37], Theorem 2.4: "Let the p-subgroup P be a p-uniqueness subgroup in the finite simple group S which belongs to one of the seven rank-unbounded families. Then the rank of S is bounded in terms of P.") about the rank-unbounded ones of the 19 known families of finite simple groups. We introduce a new scheme to describe the 19 families, the family T of types, define the rank of each type, and emphasise the rôle of Kegel covers. This part presents a unified picture of known results whose proofs are by reference. Subsequently we apply new ideas to prove the conjecture for the alternating groups. Thereupon we are remembering Kegel covers and *-sequences. Next we suggest a way 1) and a way 2) how to prove and even how to optimise Kegel's conjecture step-by-step or peu à peu which leads to Conjecture 1, Conjecture 2 and Conjecture 3 thereby unifying Sylow theory in locally finite simple groups with Sylow theory in locally finite and p-soluble groups whose joint study directs Sylow theory in (locally) finite groups. For any unexplained terminology we refer to [15]. We then continue the program begun above to optimise along the way 1) the 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 Conjecture 2 about the General Linear Groups over (commutative) locally finite fields, stating that their rank is bounded in terms of their p-uniqueness, and then break down this insight to the Special Linear Groups and the Projective Special Linear (PSL) Groups over locally finite fields. We close with suggestions for future research -> regarding the remaining rank-unbounded types (the "Classical Groups") and the way 2), -> regarding (locally) finite and p-soluble groups, and -> regarding Cauchy's and Galois' contributions to Sylow theory in finite groups. We much hope to enthuse group theorists with them. We include the predecessor research paper [15] as an Appendix.
In Part 3 of the First 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" we continue the program begun in [10] to optimise along the way 1) its beautiful 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 beautiful Conjecture 2 of [10] about the General Linear Groups over (commutative) locally finite fields, stating that their rank is bounded in terms of their p-uniqueness, and then break down this insight to the Special Linear Groups and the Projective Special Linear (PSL) Groups over locally finite fields. We close with suggestions for future research -> regarding the remaining rank-unbounded types (the "Classical Groups") and the way 2), -> regarding the (locally) finite and p-soluble groups, and -> regarding Augustin-Louis Cauchy's and Évariste Galois' contributions to Sylow theory in finite groups, which culminate in the announcement of a Second Trilogy.
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 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].
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.
Part 3 of the second Trilogy "The Strong Sylow Theorem for the Prime p in the Locally Finite Classical Groups" & "The Strong Sylow Theorem for the Prime p in Locally Finite and p-Soluble Groups" & "Augustin-Louis Cauchy's and Évariste Galois' Contributions to Sylow Theory in Finite Groups" proves for a subgroup G of the finite group H Lagrange's theorem and three group theorems by Cauchy, where the second and the third were concealed, by a unified method of proof consisting in smart arranging the elements of H resp. the cosets of G in H in a rectangle/tableau. Cauchy's third theorem requires the existence of a Sylow p-subgroup of H. These classical proofs are supplemented by modern proofs based on cosets resp. double cosets which take only a few lines. We then analyse first his well-known published group theorem of 1845/1846, for which he constructs a Sylow p-subgroup of Sn, thereby correcting a misunderstanding in the literature and introducing wreath products, and second his published group theorem of 1812/1815, which is related to theorems of Lagrange, Vandermonde and Ruffini. Subsequently we present what Galois knew about Cauchy's group theorems and about Sylow's theorems by referring to his published papers and as well to his posthumously published papers and to his manuscripts. We close with a detailed narrative of early group theory and early Sylow theory in finite groups.
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.
The series is aimed specifically at publishing peer reviewed reviews and contributions presented at workshops and conferences. Each volume is associated with a particular conference, symposium or workshop. These events cover various topics within pure and applied mathematics and provide up-to-date coverage of new developments, methods and applications.