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

Abstract

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 puniqueness subgroup of the finite simple group S which belongs to one of the 7 rankunbounded families. Then the rank of S will be bounded in terms of P ." ) about the rankunbounded 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

Review of a Trilogy about Sylow Theory in Locally Finite Groups

“…Part 1, "Characterising Locally Finite Groups Satisfying the Strong Sylow Theorem for the Prime p" of the trilogy [2][3][4] is based on the BoD-Book "Characterising locally finite groups satisfying the strong Sylow Theorem for the prime p -Revised edition" (see under ISBN 978-3-7562-3416-5), which in turn has been based on the author's quite pioneering research paper in Volume 13 of AGTA (see https://www.advgrouptheory.com/journal/Volumes/13/Flemisch. pdf and the summary in Zbl 1496.20065).…”

“…Part 2, "About the Strong Sylow Theorem for the Prime p in Simple Locally Finite Groups" of the trilogy [5] is based on Part 1. It first gives a profound overview of the structure of simple groups, and in particular of the simple locally finite groups, and reduces their Sylow theory for the prime p to an old (1987) conjecture of Professor Otto H. Kegel, contained in his second paper: "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.…”

“…Part 3, "The Strong Sylow Theorem for the Prime p in Projective Special Linear Locally Finite Groups" of the trilogy [6] continues the program begun in Part 2 to optimise along way 1), the theorem about the first type Ξ = "A n " of infinite families of finite simple groups step-by-step to further types by proving it for the second type Ξ = "A = PSL n ". It first proves the Conjecture 2 from Part 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 breaks down this insight into the Special Linear Groups and the Projective Special Linear (PSL) Groups over locally finite fields.…”