Loops and Semidirect Products

Details Loops and Semidirect Products Loops and Semidirect Products

1999-07-13

arxiv.org/abs/math/9907085v3

Comm. Algebra 28(9) (2000), pp. 4137-4164

Mathematics

Group Theory

27 pages, LaTeX2e, uses tcilatex.sty; final version; to appear in Comm. Algebra

Scientific paper

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.

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