A class of loops categorically isomorphic to Bruck loops of odd order

TL;DR

This paper introduces $\Gamma$-loops, proves their categorical equivalence to Bruck loops of odd order, and applies this to derive key theorems and answer open questions in loop theory.

Contribution

It defines $\Gamma$-loops, establishes their isomorphism with Bruck loops of odd order, and uses this to extend classical theorems and solve open problems in automorphic loops.

Findings

Categorical isomorphism between $\Gamma$-loops and Bruck loops of odd order.

Derived Odd Order, Lagrange, and Cauchy Theorems for $\Gamma$-loops.

Confirmed the existence of Hall $\pi$-subloops and Sylow $p$-subloops in commutative automorphic loops.

Abstract

We define a new variety of loops we call $\Gamma$-loops. After showing $\Gamma$-loops are power associative, our main goal will be showing a categorical isomorphism between Bruck loops of odd order and $\Gamma$-loops of odd order. Once this has been established, we can use the well known structure of Bruck loops of odd order to derive the Odd Order, Lagrange and Cauchy Theorems for $\Gamma$-loops of odd order, as well as the nontriviality of the center of finite $\Gamma$-$p$-loops ($p$ odd). Finally, we answer a question posed by Jedli\v{c}ka, Kinyon and Vojt\v{e}chovsk\'{y} about the existence of Hall $\pi$-subloops and Sylow $p$-subloops in commutative automorphic loops. By showing commutative automorphic loops are $\Gamma$-loops and using the categorical isomorphism, we answer in the affirmative.