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  "body": "\n## Definition\n\nLet $G$ be a finite group and let $K$ be a field whose [[characteristic|0F6ES3BRYP]] is a prime $p$ that divides $\\abs{G}$.\nA *modular representation* of $G$ over $K$ is a [[representation|WYGRXCG2RN]] $\\rho \\colon G \\to GL(V)$ on a finite-dimensional vector space $V$ over $K$.\nEquivalently, it is a finitely generated [[module|0NZ8JA25WK]] over the [[group algebra|MNP2HBWB5Q]] $KG$.\n*Modular representation theory* is the study of these representations and modules.\n\n## Remarks\n\nExample: $G = \\Z/p\\Z$ with generator $g$, $K = \\F_p$, and $\\rho(g) = \\begin{pmatrix} 1 \u0026 1 \\\\ 0 \u0026 1 \\end{pmatrix}$.\nThis is a representation because $\\rho(g)^p = \\begin{pmatrix} 1 \u0026 p \\\\ 0 \u0026 1 \\end{pmatrix} = I$ in characteristic $p$.\nThe line $K e_1$ is invariant, but any other line $K(a e_1 + e_2)$ is sent to $K((a+1) e_1 + e_2)$, a different line, so $V$ is not a direct sum of invariant lines.\nNon-example: representations of $G$ over $\\C$, or over a [[finite field|0ZXZ82FTX2]] whose characteristic does not divide $\\abs{G}$, are ordinary representations; there Maschke's theorem makes every representation semisimple.\nLet $K$ be a splitting field for $G$.\nBrauer showed that the number of simple $KG$-modules is the number of conjugacy classes of elements of $G$ of order prime to $p$.\n",
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