{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Example\n\nLet $\\mathbb{Z}/2\\mathbb{Z} = \\{0, 1\\}$ be the [[integers modulo $2$|03VFZ2WETT]] with addition. Let $\\{1, -1\\}$ be a group under multiplication.\nDefine $f(0) = 1$ and $f(1) = -1$. Then $f$ is a bijection.\nWe check the four products: $f(0 + 0) = 1 = 1 \\cdot 1$, $f(0 + 1) = -1 = 1 \\cdot (-1)$, $f(1 + 0) = -1 = (-1) \\cdot 1$ and $f(1 + 1) = f(0) = 1 = (-1) \\cdot (-1)$.\nThus $f$ is a homomorphism and a bijection, so $f$ is an [[isomorphism|085PND4KNT]].\n\n## Remarks\n\nBoth groups are cyclic of order $2$. This agrees with the result that [[cyclic groups of the same order are isomorphic|0736AXH2ME]].\n",
  "commit": "f7cc1f60ee7d22cafedd3ef7b979a1abe7bcb828",
  "front": {
    "illustrates": [
      "0736AXH2ME"
    ],
    "instance_of": [
      "08MG7H8JW5"
    ],
    "kind": "example",
    "msc": [
      "20A05"
    ],
    "name": "The integers modulo 2 and the group of signs are isomorphic",
    "sources": [
      {
        "locator": "Cyclic groups",
        "url": "https://en.wikipedia.org/wiki/Group_isomorphism"
      }
    ],
    "tag": "7FCN2KNWC0",
    "uses": [
      "03VFZ2WETT",
      "085PND4KNT"
    ]
  },
  "kind": "example",
  "link": "https://math2.tamnd.com/node/7FCN2KNWC0/",
  "links": [
    {
      "field": "instance_of",
      "target": "08MG7H8JW5"
    },
    {
      "field": "illustrates",
      "target": "0736AXH2ME"
    },
    {
      "field": "uses",
      "target": "03VFZ2WETT"
    },
    {
      "field": "uses",
      "target": "085PND4KNT"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/7FCN2KNWC0.md",
  "msc": [
    "20A05"
  ],
  "name": "The integers modulo 2 and the group of signs are isomorphic",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/7FC/7FCN2KNWC0.md",
  "status": null,
  "tag": "7FCN2KNWC0",
  "uri": "m2://7FCN2KNWC0",
  "url": "https://math2.tamnd.com/wiki/the-integers-modulo-2-and-the-group-of-signs-are-isomorphic/",
  "writing": {
    "tag": "7FCN2KNWC0",
    "kind": "example",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}