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

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.

Felix F. Flemisch proudly received his first degree Bacc.Math. in 1974 from the Albert-Ludwigs-Universität at lovely Freiburg im Breisgau, his degree M.Sc. in 1975 from the University of London, UK, and finally his degree Dipl.-Math. at marvellous and fabulous Freiburg i.Br. in 1985. From February 1981 until April 1985 he was quite happily affiliated to the Albert-Ludwigs-Universität Freiburg i.Br., Universitätsklinikum Freiburg, Institut für Medizinische Biometrie und Statistik (IMBI). Since May 1985 he was enthusiastically with great joy working for the telecom industry. On April 11, 1992, he married beloved Helga in beautiful Florence in Tuscany in Italy. Since October 2016 he is retired and is still resp. is again loving mathematics, in particular group theory.