On a class of power associative LCC-loops

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O.O. George
J.O. Olaleru
J.O. Adénı́ran
T.G. Jaiyéolá


Let LWPC denote the identity (xy · x) · xz = x((yx · x)z), and RWPC the mirror identity. Phillips proved that a loop satisfies LWPC and RWPC if and only if it is a WIP PACC loop. Here, it is proved that a loop Q fulfils LWPC if and only if it is a left conjugacy closed (LCC) loop that fulfils the identity (xy · x)x = x(yx · x). Similarly, RWPC is equivalent to RCC and x(x · yx) = (x · xy)x. If a loop satisfies LWPC or RWPC, then it is power associative (PA). The smallest nonassociative LWPC-loop was found to be unique and of order 6 while there are exactly 6 nonassociative LWPC-loops of order 8 up to isomorphism. Methods of construction of nonassociative LWPC-loops were developed.


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George, O., Olaleru, J., Adénı́ranJ., & JaiyéoláT. (2022). On a class of power associative LCC-loops. Extracta Mathematicae, 37(2), 185-194. https://doi.org/10.17398/2605-5686.37.2.185
Group Theory


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