# 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.

## Key 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.