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  "body": "\n## Definition\n\nLet $G$ be a finite [[group|G5S1M9]] of order $v$ with identity $e$, and let $k$ and $\\lambda$ be positive integers.\nA $(v, k, \\lambda)$ difference set in $G$ is a [[subset|0BJR9QR08E]] $D \\subseteq G$ with $k$ elements with the following property.\nEach element $g \\ne e$ of $G$ can be written as $g = d_1 d_2^{-1}$ with $d_1, d_2 \\in D$ in exactly $\\lambda$ ways.\n\n## Remarks\n\nWhen $G$ is abelian and written additively, the condition is about differences $d_1 - d_2$.\nCounting the $k(k - 1)$ pairs $(d_1, d_2)$ with $d_1 \\ne d_2$ gives $k(k-1) = \\lambda (v - 1)$.\nExample: $D = \\set{1, 2, 4}$ in $\\Z/7\\Z$.\nThe six differences are $2 - 1 = 1$, $1 - 2 = 6$, $4 - 1 = 3$, $1 - 4 = 4$, $4 - 2 = 2$, $2 - 4 = 5$, so each nonzero residue occurs once, and $D$ is a $(7, 3, 1)$ difference set.\nNon-example: $D = \\set{0, 1, 2}$ in $\\Z/7\\Z$ is not a difference set, since $1 = 1 - 0 = 2 - 1$ occurs twice while $3$ does not occur.\nIf $D$ is a $(v, k, \\lambda)$ difference set with $k \u003c v$, the translates $Dg = \\set{dg : d \\in D}$ for $g \\in G$ are the blocks of a symmetric $2$-$(v, k, \\lambda)$ [[block design|TGZATGBHTH]].\n",
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