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  "body": "\n## Definition\n\nLet $G$ be an [[abelian group|082GHNPJ53]], and let $\\mathbb{T} = \\set{z \\in \\C : \\abs{z} = 1}$ be the [[circle group|04EAFVFTF7]].\nA *character* of $G$ is a [[group homomorphism|0JYTG5K9Y4]] $\\chi \\colon G \\to \\mathbb{T}$, that is, a map with $\\chi(x + y) = \\chi(x)\\chi(y)$ for all $x, y \\in G$.\nIf $G$ is a topological group, a character of $G$ is usually required to be continuous as well.\n\n## Remarks\n\nExample: for $G = \\Z/n\\Z$ and an integer $j$, the map $\\chi_j(k) = e^{2\\pi i jk/n}$ is well defined, because $e^{2\\pi i j(k + n)/n} = e^{2\\pi i jk/n}$.\nIt is a character, since $\\chi_j(k + l) = e^{2\\pi i j(k+l)/n} = \\chi_j(k)\\chi_j(l)$.\nNon-example: the map $\\Z \\to \\C$, $k \\mapsto 2^k$, satisfies the homomorphism rule, but its values do not have absolute value $1$, so it is not a character.\nThe characters of $G$ form an abelian group under pointwise multiplication $(\\chi\\psi)(x) = \\chi(x)\\psi(x)$, with identity the trivial character $x \\mapsto 1$ and inverse $\\overline{\\chi}$.\nThis group is the character group, often written $\\widehat{G}$.\nFor a finite group every homomorphism to $\\C^\\times$ takes values in $\\mathbb{T}$, so some texts define characters as homomorphisms to $\\C^\\times$.\n",
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