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  "body": "\n## Summary\n\nA functional equation on a group is a [[functional equation|SRAN4EKWS0]] whose unknown functions are defined on a [[group|G5S1M9]].\nThe arguments of these functions are formed from the variables by the group operation and inversion.\nThe basic example is Cauchy's equation $f(xy) = f(x) f(y)$ for maps $f$ from $G$ to another group, whose solutions are exactly the [[group homomorphisms|0JYTG5K9Y4]].\nClassical equations on $\\R$, such as the d'Alembert, Jensen and quadratic equations, have analogues on arbitrary groups.\nTheir solutions are often described by homomorphisms, [[characters|H1WNVN5V7F]] and representations of the group, which links the subject to harmonic analysis.\n\n## Explanations\n\n### L1 human\n\nMany functional equations on the real line only use addition and subtraction of the variables.\nAny group has an operation and inverses, so the same equations make sense on any group.\nFor example, on the group of nonzero real numbers under multiplication, $f(xy) = f(x) + f(y)$ is solved by $f(x) = \\ln \\abs{x}$.\nOn non-commutative groups, such as groups of matrices, new phenomena appear, because $xy$ and $yx$ may differ.\n",
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