{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "D1QA6N9EWJ"
    }
  ],
  "body": "\n## Summary\n\nA difference set is a subset $D$ of a finite group with a balance property.\nEvery non-identity element of the group arises as a \"difference\" $d_1 d_2^{-1}$ of elements of $D$ the same number $\\lambda$ of times.\nThe translates of a difference set form a symmetric [[block design|4JCSS2Z1BA]] on which the group acts regularly.\nDifference sets give many projective planes, Hadamard matrices and sequences with good correlation.\nThe node [[difference set|D1QA6N9EWJ]] gives the precise definition.\n\n## Explanations\n\n### L1 human\n\nTake the numbers $1, 2, 4$ in arithmetic modulo $7$.\nThe differences between two of them are $\\pm 1$, $\\pm 2$ and $\\pm 3$, and these are each nonzero residue modulo $7$ exactly once.\nSo $\\set{1, 2, 4}$ \"sees\" every nonzero distance exactly once.\nShifting it by $0, 1, \\dots, 6$ gives seven triples that form a copy of the [[Fano plane|QQ8GW1ASZM]].\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "05B10"
    ],
    "name": "Difference set",
    "same_as": {
      "wikidata": "Q5275281",
      "wikipedia": "Difference set"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Difference_set"
      }
    ],
    "tag": "MG1KM4S1BV"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/MG1KM4S1BV/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/MG1KM4S1BV.md",
  "msc": [
    "05B10"
  ],
  "name": "Difference set",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/MG1/MG1KM4S1BV.md",
  "status": null,
  "tag": "MG1KM4S1BV",
  "uri": "m2://MG1KM4S1BV",
  "url": "https://math2.tamnd.com/wiki/difference-set/",
  "writing": {
    "tag": "MG1KM4S1BV",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}