About the Strong Sylow p-Theorem in Simple Locally Finite Groups - Part 2 of a Trilogy
Manuscript on Sylow theory in locally finite groups - Part 2 of a Trilogy
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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 , 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 .
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