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  "body": "\n## Summary\n\nThe group algebra $F[G]$ of a [[group|G5S1M9]] $G$ over a field $F$ is the vector space with basis $G$.\nIt is a [[ring|0B1261B5ED]], with the group multiplication extended linearly.\nIts modules are exactly the [[representations|WYGRXCG2RN]] of $G$ over $F$.\nSo representation theory of groups becomes module theory of a single ring.\nOver a commutative ring $R$ in place of $F$, the same construction gives the group ring $R[G]$.\n\n## Explanations\n\n### L1 human\n\nTake formal sums like $2g + 3h - k$ of group elements with coefficients in a field.\nAdd them term by term, and multiply them by expanding and using the group law, so $(ag)(bh) = ab \\, (gh)$.\nFor the cyclic group $\\set{1, t}$ of order $2$, the product $(1 + t)(1 - t)$ equals $1 - t^2 = 1 - 1 = 0$.\nThis shows the group algebra can have zero divisors even though the field has none.\n",
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