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Isotopisms of quadratic quasigroups
Allsop, Jack1
1Institut für Mathematik, Freie Universität Berlin, Germany
Algebraic Combinatorics, Volume 9 (2026) no. 1, pp. 75-94

A quasigroup is a pair $(Q, *)$ where $Q$ is a non-empty set and $*$ is a binary operation on $Q$ such that for every $(u, v) \in Q^2$ there exists a unique $(x, y) \in Q^2$ such that $u * x = v = y * u$. Let $q$ be an odd prime power, let $\mathbb{F}_q$ denote the finite field of order $q$, and let $\mathcal{R}_q$ denote the set of non-zero squares in $\mathbb{F}_q$. Let $(a, b) \in \mathbb{F}_q^2$ be such that $\lbrace ab, (a-1)(b-1)\rbrace \subseteq \mathcal{R}_q$. Let $\mathcal{Q}_{a, b}$ denote the quadratic quasigroup $(\mathbb{F}_q, *_{a, b})$ where $*_{a, b}$ is defined by

\[ x*_{a, b}y = {\left\lbrace \begin{array}{ll} x+a(y-x) & \text{if } y-x \in \mathcal{R}_q, \\ x+b(y-x) & \text{otherwise}. \end{array}\right.} \]

The operation table of a quadratic quasigroup is a quadratic Latin square. Recently, it has been determined exactly when two quadratic quasigroups are isomorphic and the automorphism group of any quadratic quasigroup has been determined. In this paper, we extend these results. We determine exactly when two quadratic quasigroups are isotopic and we determine the autotopism group of any quadratic quasigroup. In the process, we count the number of $2 \times 2$ subsquares in quadratic Latin squares.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.467
Classification: 20N05, 05B15, 20B25
Keywords: quasigroup, finite field, quadratic orthomorphism, automorphism group, autotopism group, Latin square, intercalate

Allsop, Jack  1

1 Institut für Mathematik, Freie Universität Berlin, Germany
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Allsop, Jack. Isotopisms of quadratic quasigroups. Algebraic Combinatorics, Volume 9 (2026) no. 1, pp. 75-94. doi: 10.5802/alco.467

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