Loops and Semidirect Products

Abstract

A \emph{loop} $(B,\cdot)$ is a set $B$ together with a binary operation $\cdot$ such that (i) for each $a\in B$, the left and right translation mappings $L_{a}:B\to B: x \mapsto a\cdot x$ and $R_{a}:B\to B: x \mapsto x\cdot a$ are bijections, and (ii) there exists a two-sided identity element $1\in B$. Thus loops can be thought of as "nonassociative groups". In this paper we study standard, internal and external semidirect products of loops with groups. These are generalizations of the familiar semidirect product of groups.