Finite Bruck Loops

Abstract

A loop $(X,\circ)$ is said to be a Bruck loop if it satisfies the (right) Bol identity $((z\circ x)\circ y)\circ x = z\circ ((x\circ y)\circ x)$ and the automorphic inverse property $(x\circ y)^{-1}=x^{-1}\circ y^{-1}$. If $X$ is a finite Bruck loop and $G$ is the group generated by all right translations $R(x): y\mapsto y\circ x$, then we show that $X$ and $G$ are central products $X = O^{2'}(X) * O(X)$ and $G = O^{2'}(G) * O(G)$, where $O^{2'}(X)$ ($O^{2'}(G)$) is the subloop (subgroup) generated by all 2-elements, and $O(X)$ ($O(G)$) is the largest normal subloop (subgroup) of odd order. In particular, if $X$ is solvable, then these central products are direct products. We also give a set of necessary conditions that must hold for a finite Bruck loop $X$ to be nonsolvable but have each proper section solvable; in particular, $X$ must be simple and consist of 2-elements, while the quotient of $G$ by its largest normal 2-subgroup must be isomorphic to $PGL_2(q)$, with $q=2^n+1\geq 5$.