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  "body": "\n## Summary\n\nA character of an abelian group is a group homomorphism from the group to the circle group of complex numbers of absolute value $1$.\nThe characters of a group form a group under pointwise multiplication, the character group.\nCharacters are the \"pure waves\" of Fourier analysis on groups: every reasonable function on the group is built from them.\n\n## Explanations\n\n### L1 human\n\nA character labels each element of the group by a point on the unit circle, so that adding elements multiplies the labels.\nOn the cyclic group $\\Z/n\\Z$, the map $k \\mapsto e^{2\\pi i k/n}$ is a character: going once around the group goes once around the circle.\nMore generally $k \\mapsto e^{2\\pi i jk/n}$ is a character for each $j$, and these are all of them.\nOn the real line, the maps $x \\mapsto e^{i\\xi x}$ for real $\\xi$ are the continuous characters, and they are the waves used in the Fourier transform.\n",
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