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January, 2021 Commutator theory for racks and quandles
Marco BONATTO, David STANOVSKÝ
J. Math. Soc. Japan 73(1): 41-75 (January, 2021). DOI: 10.2969/jmsj/83168316

Abstract

We adapt the commutator theory of universal algebra to the particular setting of racks and quandles, exploiting a Galois connection between congruences and certain normal subgroups of the displacement group. Congruence properties, such as abelianness and centrality, are reflected by the corresponding relative displacement groups, and the global properties, solvability and nilpotence, are reflected by the properties of the whole displacement group. To show the new tool in action, we present three applications: non-existence theorems for quandles (no connected involutory quandles of order $2^k$, no latin quandles of order $\equiv2\pmod4$), a non-colorability theorem (knots with trivial Alexander polynomial are not colorable by solvable quandles; in particular, by finite latin quandles), and a strengthening of Glauberman's results on Bruck loops of odd order.

Funding Statement

This research was partly supported by the GAČR grant 18-20123S.

Citation

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Marco BONATTO. David STANOVSKÝ. "Commutator theory for racks and quandles." J. Math. Soc. Japan 73 (1) 41 - 75, January, 2021. https://doi.org/10.2969/jmsj/83168316

Information

Received: 19 August 2019; Published: January, 2021
First available in Project Euclid: 12 December 2020

Digital Object Identifier: 10.2969/jmsj/83168316

Subjects:
Primary: 57M27
Secondary: 08A30, 20N02, 20N05

Rights: Copyright © 2021 Mathematical Society of Japan

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Vol.73 • No. 1 • January, 2021
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