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

Abstract

This Research Article 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 [44], 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 well-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 great rôle of Kegel covers: Prof. Kegel rediscovered from Prof. Philip Hall (see [46]) that an infinite simple group has a local system consisting of countably infinite simple subgroups (see [45], [46] and [44], Theorem 2.5) (and conversely) and if they are locally finite he discovered groundbreakingly that they have a Kegel cover (see [44], Theorem 2.6), that is, a nested local system {G_n} with maximal normal subgroups M_n ≤ G_n such that G_n ∩ M_n+1 = <1> so that G_n embeds into G_n+1 / M_n+1. This part presents a unified picture of known results all of 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 and the classification of simple locally finite groups according to their Kegel covers. 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 very reliably 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 Ξ = “Aⁿ ” of infinite families of finite simple groups step-by-step to further types by proving it for the second type Ξ = “A = PSL_n”. We apply new ideas to prove Conjecture 2 about the General Linear Groups over 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 to the Projective Special Linear (PSL) Groups over locally finite fields. We close with good 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 these suggestions and are ready to support and to cöordinate all related work.

It follows from our two theorems that simple locally finite groups which satisfy the Strong Sylow theorem for even one Prime p are linear and hence countable if they have a local system of countable simple subgroups each having a Kegel cover “of alternating type” or “of projective special linear type”.

We include the beautiful predecessor Research Article [15] as the First Appendix for good reasons. This Research Article had been presented as a slideshow in a Talk at IGT 2024 on April 11. We include its 16 slides as the Second Appendix. Slide 1 to Slide 12 had as well been permanently instaled during IGT 2024 as a Permanent Poster.

The Research Article consists of the following seventeen beautiful Chapters:

• Sketch of proof for Aⁿ ; ● Sketch of proof for A = PSL_n ; ①Introduction; ②Proof of Theorem 1;
  ③About Kegel covers; ④Planning future research – Part 1; ⑤Proof of Theorem 2;
  ⑥Proof of Theorem 3; ⑦Proof of Theorem 4; ⑧Planning future research – Part 2;
  ⑨The First Trilogy and The Second Trilogy and their reviews; ● Acknowledgements;
  ● Postscript, Luciano De Creszenzo, Felix F. Flemisch, Conflicts of Interest, Pablo Picasso’s La Joie de vivre ;
  ● About the author in Munich, in Freiburg i.Br., in London, in Weiden i.d.OPf., and in Florence in Tuscany in Italy;
  ● 75 References; ● Appendix 1 – Reference [15] with MR Review and Zbl Review;
  ● Appendix 2 – Talk by Felix F. Flemisch at Ischia Group Theory 2024.