Enumeration of involutory latin quandles, Bruck loops and commutative automorphic loops of odd prime power order

Abstract

There is a one-to-one correspondence between involutory latin quandles and uniquely $2$-divisible Bruck loops. Bruck loops of odd prime power order are centrally nilpotent. Using linear-algebraic approach to central extensions, we enumerate Bruck loops (and hence involutory latin quandles) of order $3^k$ for $k\le 5$, except for those loops that are central extensions of the cyclic group of order $3$ by the elementary abelian group of order $3^4$. Among the constructed loops there is a Bruck loop of order $3^5$ whose associated $\Gamma$-loop is not a commutative automorphic loop. We independently enumerate commutative automorphic loops of order $3^k$ for $k\le 5$, with the same omission as in the case of Bruck loops.