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In the paper “Good involutions of conjugation subquandles” [LT25], I posed a question about involutions in certain central subgroups of nonabelian groups. This very short blogpost answers that question in the positive. Our example is the central product \((D_4\times C_4)/C_2\), which has order 16.
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In this blogpost, we provide infinitely many counterexamples to a conjecture of Bardakov and Fedoseev [BF24] about minimal generating subsets of core quandles of groups. Specifically, we show that \(\mathrm{rank}(\mathrm{Core}(D_n))=4\) for all \(n\geq 2\). The proof is elementary and eschews the more powerful results of Bergman [Be21].
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In this blogpost, we solve two open problems posed by Bardakov and Elhamdadi [BE26] in the theory of quandle rings. In particular, non-medial commutative quandles obstruct a conjectural structure theorem of [op. cit.]. However, assuming mediality makes the conjecture hold; we deduce this from an equivalence of categories between commutative (resp. Latin) medial quandles and affine modules over the ring of dyadic rationals (resp. integral Laurent polynomials in \(s\) and \(1-s\)).