Characterising locally finite groups satisfying the strong Sylow Theorem for the prime p – Mathematics edition – Second edition
DOI:
https://doi.org/10.47363/JMCA/2026(5)239Keywords:
conjugacy of maximal (i.e. Sylow) p-subgroups, singular (Sylow) p-subgroup, locally 41 finite group satisfying the (strong) Sylow Theorem for the prime p (or the [strong] Sylow 42 p-Theorem) (G Syl-p), finite simple group, split sequence of finite p-perfect subgroups, (very) 43 good Sylow p-subgroup, p-uniqueness subgroup, singular Sylow p-subgroup, minimal p-unique 44 subgroup, (proper) Sylow p-intersection, the (numerical) Sylow p-invariants 1) p-uniqueness 45 and 2) Sylow p-intersection, Sylow-separated sequence of p-subgroups, maximum conditions 46 (for I pG and for the Sylow-separated sequences of p-subgroups), locally finite group G 47 satisfying the minimum condition for p-subgroups (G Min-p), p-blank of G in HAbstract
We turn a question of Otto H. Kegel [1] into a theorem [2]: If every subgroup S of the locally finite group G contains a finite p-subgroup which is singular in S (see below), then G satisfies the strong Sylow Theorem for the prime p (see below). Since the converse is also true [1; 2], this characterises the locally finite groups which satisfy the strong Sylow Theorem for the prime p.
The proof is a rather old result, namely the Charakterisierungssatz of the author’s Diplomarbeit of 1984 [2]. It is based upon the concepts of good p-subgroups [3] and strongly local – or, as we say, very good – p-subgroups [4] of Andrew Rae.
To obtain the correspondence with Kegel’s work we first show that a finite p-subgroup is singular in the locally finite group G if and only if it is a p-uniqueness subgroup of G, that is, it is contained in a unique maximal p-subgroup of G. We then proceed on the whole as in our Diplomarbeit.
However, we do not publish the proof in its original form since the Diplomarbeit was written in German. Instead we translated all employed parts into English, thereby introducing a number of corrections, present the Charakterisierungssatz as 2.8 HAUPTSATZ, and summarise then the result as Theorem IV.
Subsequently we present a few new concepts and results for Sylow theory in (locally) finite groups and a comparative outline of Kegel’s previous work [8] with two open questions. We close with personal dedications, background information and deep endnotes which introduce further new concepts.
The publication has the following seven chapters: Introduction; Good Sylow p-subgroups and p-uniqueness subgroups; Basic theorems of Sylow theory in locally finite groups and our Charakterisierungssatz; Novel concepts and results for Sylow theory in (locally) finite groups and comparative outline of Kegel’s previous work [8]; References; About the author & background information and annotations; Endnotes.
