On Bruck Loops of 2-power Exponent

TL;DR

This paper investigates the structure of Bruck loops of 2-power exponent using group theoretic methods, under specific assumptions about simple groups, and provides foundational information for further classification and analysis.

Contribution

It offers new insights into the groups associated with Bruck loops of 2-power exponent under certain simple group assumptions.

Findings

Determined groups associated with Bruck loops of 2-power exponent.

Provided group-theoretic framework for studying Bol and Bruck loops.

Supported future classification of finite Bruck loops.

Abstract

The goal of this paper is two-fold. First we provide the information needed to study Bol, $A_r$ or Bruck loops by applying group theoretic methods. This information is used in this paper as well as in [BS3] and in [S]. Moreover, we determine the groups associated to Bruck loops of 2-power exponent under the assumption that every nonabelian simple group $S$ is either passive or isomorphic to $\PSL_2(q)$, $q-1 \ge 4$ a $2$-power. In a separate paper it is proven that indeed every nonabelian simple group $S$ is either passive or isomorphic to $\PSL_2(q)$, $q-1 \ge 4$ a $2$-power [S]. The results obtained here are used in [BS3], where we determine the structure of the groups associated to the finite Bruck loops.