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        "quote": "> [!exercise] <a class=\"object-permalink\" href=\"#problem-2-16\">习题 2.16</a> \n>\n> <u id=\"concept-2-mobius-transformation\" class=\"concept-term\" data-concept-identity=\"urn:uuid:6328536c-e4a4-4277-b275-0a8a7146d152\" data-concept-route=\"/concept-identities#concept-6328536c-e4a4-4277-b275-0a8a7146d152\">莫比乌斯变换(Möbius transformation)</u>是复映射\n>\n> $$\n> z\\mapsto z'=\\frac{az+b}{cz+d},\\qquad a,b,c,d\\in\\mathbb{C},\\quad ad-bc=1.\n> $$\n>\n> (a) 证明这些映射是扩充复平面（包含点 $z=\\infty$）上的一一到上变换，并写出分别由常数 $(a,b,c,d)$ 和 $(a',b',c',d')$ 确定的任意两个变换的复合。\n>\n> (b) 证明它们组成一个群，称为<u id=\"concept-2-mobius-group\" class=\"concept-term\" data-concept-identity=\"urn:uuid:861e1ce9-8826-4d7f-b756-fead75d051ac\" data-concept-route=\"/concept-identities#concept-861e1ce9-8826-4d7f-b756-fead75d051ac\">莫比乌斯群(Möbius group)</u>。\n>\n> (c) 证明把幺模矩阵 $\\begin{pmatrix}a&b\\\\c&d\\end{pmatrix}$ 映到上述莫比乌斯变换的映射 $\\mu$，是从 $SL(2,\\mathbb{C})$ 到莫比乌斯群的同态，其核为 $\\{\\mathsf{I},-\\mathsf{I}\\}$。也就是说，莫比乌斯群同构于 $SL(2,\\mathbb{C})/\\mathbb{Z}_2$。",
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