{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $G$ be a [[topological group|92700QAB5Y]] with identity $e$.\nThe group $G$ has *no small subgroups* if some [[neighbourhood|0ZCY0Z80B6]] $U$ of $e$ has this property.\nThe only [[subgroup|0X1JXT47GA]] $H$ of $G$ with $H \\subseteq U$ is $H = \\set{e}$.\n\n## Remarks\n\nExample: $G = (\\R, +)$ with $U = (-1, 1)$.\nIf $H \\subseteq U$ is a subgroup and $x \\in H$ with $x \\ne 0$, then $nx \\in H$ for every integer $n$, and $\\abs{nx} \\ge 1$ for $n \\ge 1/\\abs{x}$, so $nx \\notin U$.\nHence $H = \\set{0}$.\nThe same argument with the exponential map shows that every [[Lie group|7JBRGCVJRB]] has no small subgroups.\n\nNon-example: the group $\\Z_p$ of [[p-adic integers|V6TJAFZ4MD]].\nEvery neighbourhood of $0$ contains a ball $p^n \\Z_p$ for some $n$, and $p^n \\Z_p$ is a subgroup different from $\\set{0}$.\n\nThe property is often abbreviated NSS.\nBy the Gleason–Yamabe theorem, a locally compact group has no small subgroups if and only if it is a Lie group.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "HPMAJV2RDY",
    "kind": "definition",
    "msc": [
      "22A05",
      "22E05",
      "22D05"
    ],
    "name": "No small subgroups",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/No_small_subgroup"
      }
    ],
    "tag": "V3N4HX0GRP",
    "uses": [
      "92700QAB5Y",
      "0ZCY0Z80B6",
      "0X1JXT47GA"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/V3N4HX0GRP/",
  "links": [
    {
      "field": "defines",
      "target": "HPMAJV2RDY"
    },
    {
      "field": "uses",
      "target": "92700QAB5Y"
    },
    {
      "field": "uses",
      "target": "0ZCY0Z80B6"
    },
    {
      "field": "uses",
      "target": "0X1JXT47GA"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/V3N4HX0GRP.md",
  "msc": [
    "22A05",
    "22E05",
    "22D05"
  ],
  "name": "No small subgroups",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/V3N/V3N4HX0GRP.md",
  "status": null,
  "tag": "V3N4HX0GRP",
  "uri": "m2://V3N4HX0GRP",
  "url": "https://tamnd.github.io/math2/wiki/no-small-subgroups-definition/",
  "writing": {
    "tag": "V3N4HX0GRP",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}