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Diassociativity in Conjugacy Closed Loops

Michael K. Kinyon·Kenneth Kunen·J. D. Phillips
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Abstract

Let $Q$ be a conjugacy closed loop, and $N(Q)$ its nucleus. Then $Z(N(Q))$ contains all associators of elements of $Q$. If in addition $Q$ is diassociative (i.e., an extra loop), then all these associators have order 2. If $Q$ is power-associative and $|Q|$ is finite and relatively prime to 6, then $Q$ is a group. If $Q$ is a finite non-associative extra loop, then $16 \mid |Q|$.

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Michael K. Kinyon, Kenneth Kunen, J. D. Phillips. 2002-09-20. Diassociativity in Conjugacy Closed Loops. https://arxiv.org/abs/math/0209279

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