A special class of Moufang loops

Loops satisfying the identical relation $(xy)(zx) = [x(yz)]x$ are known as Moufang loops. In the present paper considered is a class of loops satisfying the identical relation \[ (1)\qquad (xy)(z{x^\lambda }) = [x(yz)]{x^\lambda },\] where ${x^\lambda }$ is the image of $x$ under some mapping $\lambda$ of the loop into itself. Loops satisfying (1) are called $M$-loops. If $\lambda :x \to {x^k},k$ an integer, (1) is called an ${M_k}$-law. It is shown that every $M$-loop is Moufang, and every ${x^\delta } = {x^{ - 1}} \cdot {x^\lambda }$ belongs to the nucleus. Furthermore, if a loop satisfies (1) so do all its loop-isotopes.