First Trilogy about Sylow Theory in Locally Finite Groups

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Part 1 of the Trilogy is derived from the BoD-Book focused on characterizing locally finite groups that satisfy the strong Sylow Theorem for the prime p. The first edition of Part 1 removes highlights from the revised edition, adds 14 pages to the AGTA paper, and includes references as appendices while acknowledging Professor Otto H. Kegel's contributions to the Ischia Group Theory conference. The second edition features uniform page numbering, improved content, additional pages, and a Table of Contents. Part 2 is based on the author's research regarding the Strong Sylow Theorem for simple locally finite groups, providing an overview and reducing Sylow theory for the prime p to a conjecture by Kegel concerning rank-unbounded groups among the 19 known families of finite simple groups. This part introduces a new scheme for describing these families, defines the rank of each type, and highlights the significance of Kegel covers. It presents a unified perspective on existing results and applies new ideas to prove the conjecture for alternating groups. A step-by-step plan for proving the conjecture is suggested, linking Sylow theory in locally finite simple groups with that in locally finite and p-soluble groups. In Part 3, the program continues by optimizing the theorem for infinite families of finite simple groups, proving it for the second type, and concluding with future research suggestions regarding remaining rank-unbounded t