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  "body": "\n## Example\n\nLet $G = (\\mathbb{Z}, +)$ be the [[integers|0C4JS9RF5M]] with addition. Define $d : G \\to G$ by $d(u) = 2u$.\nThen $d(u + v) = 2u + 2v = d(u) + d(v)$, so $d$ is a homomorphism.\nIf $2u = 0$, then $u = 0$. Thus the kernel is $\\{0\\}$, and $d$ is [[injective|0PVG1CRAJN]].\nThe image of $d$ is the set of even integers. Thus $1$ is not in the image, and $d$ is not [[surjective|0XKH881BZ7]].\nThus $d$ is not an [[isomorphism|085PND4KNT]].\n\n## Remarks\n\nThe image $2\\mathbb{Z}$ is isomorphic to $\\mathbb{Z}$. Thus $d$ is an isomorphism from $\\mathbb{Z}$ onto $2\\mathbb{Z}$, but not onto $\\mathbb{Z}$.\n",
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