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  "body": "\n## Definition\n\nLet $G$ be a [[group|G5S1M9]] and let $F$ be a [[field|04C5ZBG1WT]].\nThe *group algebra* $F[G]$ is the set of functions $a \\colon G \\to F$ with $a(g) \\ne 0$ for only finitely many $g \\in G$, written as finite formal sums $\\sum_{g \\in G} a_g \\, g$.\nIt is a [[vector space|000AYPA9AJ]] over $F$ under termwise addition and scalar multiplication, with basis $G$.\nIts multiplication is\n$$\\Big(\\sum_{g} a_g \\, g\\Big)\\Big(\\sum_{h} b_h \\, h\\Big) = \\sum_{k \\in G} \\Big(\\sum_{gh = k} a_g b_h\\Big) k.$$\n\n## Remarks\n\nWith this product $F[G]$ is an associative [[ring|0B1261B5ED]] with identity $1_G$, and an algebra over $F$ of dimension $\\abs{G}$.\nIt is commutative if and only if $G$ is abelian.\nExample: for $G = \\set{1, t}$ cyclic of order $2$ and $F = \\Q$, the elements $e = \\tfrac{1}{2}(1 + t)$ and $f = \\tfrac{1}{2}(1 - t)$ satisfy $e^2 = e$, $f^2 = f$, $ef = 0$ and $e + f = 1$, since $t^2 = 1$.\nSo $\\Q[G] \\cong \\Q \\times \\Q$ as rings.\nNon-example: for infinite $G$, the set of all functions $G \\to F$ is not the group algebra, since it fails the finiteness condition.\nFor $G = \\Z$ the product formula then breaks down, as the inner sum $\\sum_{gh = k} a_g b_h$ can be an infinite sum.\nA [[representation|WYGRXCG2RN]] of $G$ on $V$ is the same as a left $F[G]$-module structure on $V$.\nFor a commutative ring $R$, the same definition gives the group ring $R[G]$.\n",
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