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  "body": "\n## Definition\n\nLet $G$ and $H$ be [[groups|G5S1M9]].\nA group homomorphism from $G$ to $H$ is a map $\\varphi \\colon G \\to H$ such that $\\varphi(ab) = \\varphi(a) \\varphi(b)$ for all $a, b \\in G$.\nHere the product $ab$ is taken in $G$ and the product $\\varphi(a) \\varphi(b)$ is taken in $H$.\n\n## Remarks\n\nThe map $n \\mapsto 2^n$ is a homomorphism from $(\\mathbb{Z}, +)$ to $(\\mathbb{Q}^\\times, \\cdot)$.\nThe map $x \\mapsto e^x$ is a homomorphism from $(\\mathbb{R}, +)$ to the positive real numbers with multiplication.\nFor each pair of groups, the map that sends each element of $G$ to the identity of $H$ is a homomorphism, the trivial homomorphism.\nThe composite of two homomorphisms is a homomorphism.\n\n## Explanations\n\n### L1 human\n\nA homomorphism translates one group into another so that products go to products.\nIt does not matter if you multiply first and then translate, or translate first and then multiply.\nThe logarithm turns multiplication of positive numbers into addition, and this is why it is a homomorphism.\n",
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