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  "body": "\n## Definition\n\nLet $G$ and $H$ be [[groups|G5S1M9]] and let $\\varphi \\colon G \\to H$ be a [[group homomorphism|0JYTG5K9Y4]].\nThe image of $\\varphi$ is the set $\\operatorname{im} \\varphi = \\varphi(G) = \\{\\varphi(g) : g \\in G\\}$.\n\n## Remarks\n\nThe image is a [[subgroup|0X1JXT47GA]] of $H$, because $\\varphi(a) \\varphi(b)^{-1} = \\varphi(ab^{-1})$.\nThe homomorphism $\\varphi$ is surjective exactly when $\\operatorname{im} \\varphi = H$.\nThe image of the map $n \\mapsto 2^n$ from $\\mathbb{Z}$ to $\\mathbb{Q}^\\times$ is the set of integer powers of $2$.\n\n## Explanations\n\n### L1 human\n\nThe image is the part of the target group that the homomorphism actually reaches.\nIt is always a group of its own, inside the target.\nThe [[first isomorphism theorem|0CXXJNG5B6]] says that the image looks like a quotient of the source group.\n",
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