An “extra” law for characterizing Moufang loops

Abstract:

Let $(G, \cdot )$ be any loop and let $\lambda , \delta , \alpha$ be mappings of G into G so that $x\lambda = x \cdot x\delta = x(x\alpha \cdot x)$ for all $x \in G$. It is shown that the following conditions are equivalent: (a) $(xy \cdot z)x\alpha = x(y(z \cdot x\alpha ))$ for all $x,y,z \in G$, (b) $(G, \cdot )$ is Moufang and $x\delta$ is in the nucleus of $(G, \cdot )$ for all $x \in G$, (c) $(xy)(z \cdot x\lambda ) = (x \cdot yz)x\lambda$ for all $x,y,z \in G$. In particular, a loop $(G, \cdot )$ is extra in that $(xy \cdot z)x = x(y \cdot zx)$ for all $x,y,z \in G$ if and only if it satisfies the ${M_3}$-law in that $(xy)(z \cdot {x^3}) = (x \cdot yz){x^3}$ for all $x,y,z \in G$.