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        "quote": "上述例子中的所有群都是交换群。非交换群最常见的例子是一类叫做<u id=\"concept-2-transformation-group\" class=\"concept-term\" data-concept-identity=\"urn:uuid:a0008efe-3fef-4617-9ff0-159a32b75bde\" data-concept-route=\"/concept-identities#concept-a0008efe-3fef-4617-9ff0-159a32b75bde\">变换群(transformation group)</u>的群。我们回顾<a href=\"../chapter-01/04-mappings#section-1-4\" class=\"cross-reference\" id=\"xref-02-transformation-and-permutation-groups-1\" data-reference=\"section-1-4\">第 1.4 节</a>，集合 $X$ 的变换是一个一一对应且到的映射 $g:X\\to X$。映射 $g$ 有一个反映射 $g^{-1}:X\\to X$，使得 $g^{-1}\\circ g=g\\circ g^{-1}=\\mathrm{id}_X$ 让两个变换 $g$ 和 $h$ 的乘积定义为它们的函数组合 $gh=g\\circ h$",
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        "quote": "群常以<span id=\"concept-1-transformation-group\" class=\"block-anchor\"></span><u class=\"concept-term\" data-concept-id=\"concept-1-transformation-group\" data-concept-role=\"discussion\" data-concept-identity=\"urn:uuid:a0008efe-3fef-4617-9ff0-159a32b75bde\" data-concept-route=\"/concept-identities#concept-a0008efe-3fef-4617-9ff0-159a32b75bde\" id=\"first-134ed74213a223ada75d-zh\">变换群(transformation group)</u>的形式出现，也就是说，群元素作为某个特定几何对象上的变换而作用。以上述三维旋转群为例，这些旋转就是 $\\mathbf R^3$ 上的[<u class=\"concept-term\" data-concept-reference=\"concepts/linear-algebra/linear-map\" data-concept-identity=\"urn:uuid:9d1874b8-6c8c-465e-849a-f2ddc348d3d0\" data-concept-route=\"/concept-identities#concept-9d1874b8-6c8c-465e-849a-f2ddc348d3d0\" id=\"first-5e64326d1ddbf841b8da-zh\">线性变换(linear transformation)</u>](../../../../../concepts/linear-algebra/linear-map)。一般而言：",
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