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  "body": "\n## Definition\n\nLet $G$ be a [[group|G5S1M9]] with identity $e$, and let $Y$ be a set.\nA *functional equation on $G$* is a [[functional equation|SRAN4EKWS0]] for one or more unknown [[functions|0EPAT51DXQ]] $f_1, \\dots, f_m \\colon G \\to Y$ in which:\n\n1. the variables $x_1, \\dots, x_k$ range over all of $G$;\n2. each unknown function is evaluated at words in $x_1, \\dots, x_k$, their inverses and fixed elements of $G$;\n3. the two sides of the equation are combined from these values by fixed operations on $Y$.\n\nA solution is a tuple $(f_1, \\dots, f_m)$ for which the equation holds for all $x_1, \\dots, x_k \\in G$.\n\n## Remarks\n\nExample: for $G$ a group and $Y = \\C$, the equation $f(xy) + f(xy^{-1}) = 2 f(x) f(y)$ is d'Alembert's equation on $G$.\nIt has the solution $f(x) = (\\chi(x) + \\chi(x)^{-1})/2$ for every homomorphism $\\chi \\colon G \\to \\C^{\\times}$, as a direct expansion shows.\nNon-example: the equation $f(f(x)) = x$ for $f \\colon G \\to G$ does not fit the definition.\nThere the unknown function is evaluated at a value of itself, not at a word in the variables, so it is an iterative equation.\nFor $G = (\\R, +)$, this setting includes the classical equations of Cauchy, d'Alembert and Pexider, and Jensen's equation in the equivalent form $f(x + y) + f(x - y) = 2 f(x)$.\nSolutions on $G$ are often required to be continuous or measurable when $G$ is a topological group.\n",
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