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  "body": "\n## Definition\n\nLet $G$ and $H$ be [[groups|G5S1M9]].\nA group isomorphism from $G$ to $H$ is a [[group homomorphism|0JYTG5K9Y4]] $\\varphi \\colon G \\to H$ that is a bijection.\nThe groups $G$ and $H$ are isomorphic, written $G \\cong H$, when there is an isomorphism from $G$ to $H$.\n\n## Remarks\n\nThe inverse map of an isomorphism is again a homomorphism, thus an isomorphism.\nIsomorphism is an equivalence relation on groups.\nThe exponential map is an isomorphism from $(\\mathbb{R}, +)$ to the positive real numbers with multiplication.\nIsomorphic groups have the same order, and corresponding elements have the same order.\n\n## Explanations\n\n### L1 human\n\nTwo groups are isomorphic when they are the same group with different names for the elements.\nAn isomorphism is a dictionary that translates the names and keeps all products correct.\nFor example, the rotations of a triangle and the integers modulo $3$ under addition are isomorphic.\n",
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