The finite Bruck Loops

TL;DR

This paper characterizes the structure of finite Bruck loops, showing they decompose into products involving odd order loops, 2-power order loops, or loops related to specific groups, and derives classical group theorems for loops.

Contribution

It provides a detailed structural classification of finite Bruck loops, extending classical theorems to this non-associative setting.

Findings

Finite Bruck loops decompose into products involving specific types of loops.

Versions of Sylow's, Lagrange's, and Hall's Theorems are established for loops.

Examples of loops related to groups PSL_2(q) with q=9 or Fermat primes are shown.

Abstract

We continue the work by Aschbacher, Kinyon and Phillips [AKP] as well as of Glauberman [Glaub1,2] by describing the structure of the finite Bruck loops. We show essentially that a finite Bruck loop $X$ is the direct product of a Bruck loop of odd order with either a soluble Bruck loop of 2-power order or a product of loops related to the groups $PSL_2(q)$, $q= 9$ or $q \geq 5$ a Fermat prime. The latter possibillity does occur as is shown in [Nag1, BS]. As corollaries we obtain versions of Sylow's, Lagrange's and Hall's Theorems for loops.