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  "body": "\n## Statement\n\nLet $(Q, \\cdot)$ be a quasigroup.\nThen $(Q, \\cdot)$ is [[medial|VZHN6GA3TA]], that is, $(x \\cdot y) \\cdot (u \\cdot v) = (x \\cdot u) \\cdot (y \\cdot v)$ for all $x, y, u, v \\in Q$, if and only if there exist:\n\n1. an [[abelian group|082GHNPJ53]] structure $(Q, +)$ on the same set;\n2. two [[automorphisms|DSQ17TNW1C]] $\\varphi$ and $\\psi$ of $(Q, +)$ with $\\varphi \\circ \\psi = \\psi \\circ \\varphi$;\n3. an element $c \\in Q$;\n\nsuch that $x \\cdot y = \\varphi(x) + \\psi(y) + c$ for all $x, y \\in Q$.\n\n## Remarks\n\nThe converse direction is a direct computation.\nWith $x \\cdot y = \\varphi(x) + \\psi(y) + c$, the product $(x \\cdot y) \\cdot (u \\cdot v)$ equals $\\varphi^2(x) + \\varphi\\psi(y) + \\psi\\varphi(u) + \\psi^2(v) + \\varphi(c) + \\psi(c) + c$.\nSwapping $y$ and $u$ gives the same value because $\\varphi\\psi = \\psi\\varphi$, and the operation is a quasigroup because $\\varphi$ and $\\psi$ are bijective.\nFor the main direction, one fixes an element of $Q$ and defines a new addition on $Q$ from the quasigroup operation by an isotopy.\nMediality forces this addition to be associative and commutative, and the quasigroup operation then becomes affine in each argument.\n\nExample: on $\\Z/5\\Z$ the operation $x \\cdot y = 2x + 3y$ is a medial quasigroup, with $\\varphi(x) = 2x$, $\\psi(y) = 3y$ and $c = 0$.\nExample: on $\\Z/3\\Z$ the operation $x \\cdot y = -x - y$ is a medial quasigroup, with $\\varphi = \\psi = -\\id$ and $c = 0$.\n\nThe quasigroup hypothesis is needed.\nThe commutative semigroup $(\\N, \\cdot)$ is medial, but it is not of this form.\nIn an operation $\\varphi(x) + \\psi(y) + c$ each map $y \\mapsto x \\cdot y$ is a bijection, while in $(\\N, \\cdot)$ the map $y \\mapsto 0 \\cdot y$ is constant.\n\n## Explanations\n\n### L1 human\n\nA quasigroup is a set with an operation whose \"multiplication table\" is a Latin square: every element appears exactly once in each row and each column.\nThe theorem says that such an operation obeys the medial law only when it comes from an abelian group in a simple way.\nOne twists each argument by a symmetry of the group, adds, and shifts by a constant.\nSo a single identity forces the whole structure to be essentially linear.\n\n### L2 human\n\nThe theorem is the main structure result for medial quasigroups.\nIt reduces questions about medial quasigroups to questions about abelian groups with two commuting automorphisms.\nIt is also called the Bruck–Murdoch–Toyoda theorem.\nIf the medial quasigroup is moreover idempotent, so that $x \\cdot x = x$, then putting $x = 0$ gives $c = 0$, and then $\\varphi + \\psi = \\id$.\nSuch quasigroups are the affine combinations $x \\cdot y = \\varphi(x) + (\\id - \\varphi)(y)$, as in the midpoint operation $(x + y)/2$ on $\\Q$.\n",
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  "name": "Toyoda–Bruck theorem",
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