Semiautomorphic Inverse Property Loops

Abstract

We define a variety of loops called semiautomorphic, inverse property loops that generalize Moufang and Steiner loops. We first show an equivalence between a previously studied variety of loops. Next we extend several known results for Moufang and Steiner loops. That is, the commutant is a subloop and if $a$ is in the commutant, then $a^2$ is a Moufang element, $a^3$ is a $c$-element and $a^6$ is in the center. Finally, we give two constructions for semiautomorphic inverse property loops based on Chein's and de Barros and Juriaans' doubling constructions.

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