About The Strong Sylow Theorem For The Prime P In Simple Locally Finite Groups Part 2 Of A Trilogy
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About the Strong Sylow Theorem for the Prime p in Simple Locally Finite Groups - Part 2 of a Trilogy
Author: Dipl.-Math. Felix Flemisch
language: en
Publisher: BoD – Books on Demand
Release Date: 2023-03-30
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].
About the Strong Sylow p-Theorem in Simple Locally Finite Groups - Part 2 of a Trilogy
Author: Dipl.-Math. Felix F. Flemisch
language: en
Publisher: BoD – Books on Demand
Release Date: 2023-11-22
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].
The Strong Sylow Theorem for the Prime p in Simple Locally Finite Groups
Author: Dipl.-Math. Felix F. Flemisch
language: en
Publisher: BoD – Books on Demand
Release Date: 2024-02-26
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.