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About the Strong Sylow Theorem for the Prime p in Simple Locally Finite Groups

Volume 7, Issue 6, December 2022    |    PP. 95-118    |PDF (3080 K)|    Pub. Date: November 22, 2022
DOI: 10.54647/mathematics11359    15 Downloads     177 Views  

Felix F. Flemisch, Mitterweg 4e, 82211 Herrsching a. Ammersee, Bavaria, Germany

We first give a quite 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 famous conjecture of Prof. Otto H. Kegel ( see [16], Theorem 2.4: “ Let P be a p-uniqueness subgroup of the finite simple group S which belongs to one of the 7 rank-unbounded families. Then the rank of S will be 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 rather complete 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 new ideas to prove the conjecture for the alternating groups.
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 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].

simple group; nested local system; family T of types of known finite simple groups; simple locally finite group of type #; Kegel cover; *-sequence; Kegel sequence; rank of a locally finite simple group; locally finite group satisfying the strong Sylow Theorem for the prime p; p-uniqueness subgroup; conjugacy class; P-isomorphic P-orbit; minimal p-unique subgroup; the (numerical) Sylow p-invariant p-uniqueness ap; p-length of a p-soluble finite group; Hall-Higman Theory

Cite this paper
Felix F. Flemisch, About the Strong Sylow Theorem for the Prime p in Simple Locally Finite Groups, SCIREA Journal of Mathematics. Vol. 7 , No. 6 , 2022 , pp. 95 - 118 . https://doi.org/10.54647/mathematics11359

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