Uniquely 2-divisible Bol loops

TL;DR

This paper explores the properties of Bol and Bruck loops, demonstrating the existence of finite Bol loops with trivial centers and constructing infinite simple Bruck loops, thereby addressing the Burnside problem.

Contribution

It introduces new examples of Bol and Bruck loops with specific properties, providing counterexamples to longstanding conjectures.

Findings

Existence of finite Bol loops of odd prime exponent with trivial center

Construction of infinite simple Bruck loops for large primes

Negative resolution of the Burnside problem for Bruck loops

Abstract

Although any finite Bol loop of odd prime exponent is solvable, we show there exist such Bol loops with trivial center. We also construct finitely generated, infinite, simple Bruck loops of odd prime exponent for sufficiently large primes. This shows that the Burnside problem for Bruck loops has a negative answer.