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        "quote": "\n\n> [!definition] Definition · <span class=\"definition-name\">Group action on a set</span>\n> <span id=\"source-block-1066\" class=\"block-anchor\"></span>\n>\n> <span class=\"author-definition-label\" aria-hidden=\"true\">Definition (<u class=\"concept-term\" data-concept-id=\"concept-1-group-action-on-a-set\" data-concept-identity=\"urn:uuid:be2d8436-d3dc-4871-a029-122366c0f690\" data-concept-route=\"/concept-identities#concept-be2d8436-d3dc-4871-a029-122366c0f690\">Group action on a set</u>). </span>An action of a group $G$ on a set $M$ is given by a map\n>\n> <span id=\"source-block-1067\" class=\"block-anchor\"></span>\n>\n> $$\n> (g, x) \\in G \\times M \\to g \\cdot x \\in M\n> $$\n>\n> <span id=\"source-block-1068\" class=\"block-anchor\"></span>\n>\n> that takes a pair $( g , x )$ of a group element $g \\in G$ and an element $x \\in M$ to another element $g \\cdot x \\in M$ such that\n>\n> <span id=\"source-block-1069\" class=\"block-anchor\"></span>\n>\n> <span id=\"equation-1-2\" class=\"block-anchor\"></span>\n>\n> $$\n> g _ {1} \\cdot (g _ {2} \\cdot x) = (g _ {1} g _ {2}) \\cdot x\\tag{1.2}\n> $$\n>\n> <span id=\"source-block-1070\" class=\"block-anchor\"></span>\n>\n> and\n>\n> <span id=\"source-block-1071\" class=\"block-anchor\"></span>\n>\n> $$\n> e \\cdot x = x\n> $$\n>\n> <span id=\"source-block-1072\" class=\"block-anchor\"></span>\n>\n> where $e$ is the identity element of $G$\n",
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        "quote": "\n> [!definition] 定义 · <span class=\"definition-name\"><span id=\"concept-1-group-action-on-a-set\" class=\"block-anchor\"></span><u class=\"concept-term\" data-concept-id=\"concept-1-group-action-on-a-set\" data-concept-identity=\"urn:uuid:be2d8436-d3dc-4871-a029-122366c0f690\" data-concept-route=\"/concept-identities#concept-be2d8436-d3dc-4871-a029-122366c0f690\" id=\"first-5dc3c5d27aaa22db3338-zh\">群在集合上的作用(group action on a set)</u></span>\n> <span id=\"source-block-1066\" class=\"block-anchor\"></span>\n>\n> <span class=\"author-definition-label\" aria-hidden=\"true\">定义（群在集合上的作用）。</span>群 $G$ 在集合 $M$ 上的作用由一个映射给出\n>\n> <span id=\"source-block-1067\" class=\"block-anchor\"></span>\n>\n> $$\n> (g, x) \\in G \\times M \\to g \\cdot x \\in M\n> $$\n>\n> <span id=\"source-block-1068\" class=\"block-anchor\"></span>\n>\n> 将一个<u class=\"concept-term\" data-concept-role=\"discussion\" data-concept-identity=\"urn:uuid:e5a0d3c6-87bc-459a-b42e-e7d38b8d1601\" data-concept-route=\"/concept-identities#concept-e5a0d3c6-87bc-459a-b42e-e7d38b8d1601\" id=\"first-2c46f30ae6e202d6ded9-zh\">有序对(ordered pair)</u> $(g,x)$（其中 $g\\in G$ 是群元素，$x\\in M$ 是集合元素）映为另一个元素 $g\\cdot x\\in M$，使得\n>\n> <span id=\"source-block-1069\" class=\"block-anchor\"></span>\n>\n> <span id=\"equation-1-2\" class=\"block-anchor\"></span>\n>\n> $$\n> g _ {1} \\cdot (g _ {2} \\cdot x) = (g _ {1} g _ {2}) \\cdot x\\tag{1.2}\n> $$\n>\n> <span id=\"source-block-1070\" class=\"block-anchor\"></span>\n>\n> 和\n>\n> <span id=\"source-block-1071\" class=\"block-anchor\"></span>\n>\n> $$\n> e \\cdot x = x\n> $$\n>\n> <span id=\"source-block-1072\" class=\"block-anchor\"></span>\n>\n> 其中 $e$ 是 $G$ 的恒等元\n",
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