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        "quote": "> [!example] <a class=\"object-permalink\" href=\"#example-2-5\">例 2.5</a> \n>\n> 如果一个群 $G$ 有一个元素 $a$，使得它的幂 $\\left\\{a, a^2, a^3, \\ldots\\right\\}$ 遍历其所有元素，则称 $G$ 为<u id=\"concept-2-cyclic-group\" class=\"concept-term\" data-concept-identity=\"urn:uuid:ff7730a6-70b3-41c7-82df-49ecbed5b2ad\" data-concept-route=\"/concept-identities#concept-ff7730a6-70b3-41c7-82df-49ecbed5b2ad\">循环群(cyclic group)</u>，而 $a$ 称为群的<u id=\"concept-2-generator\" class=\"concept-term\" data-concept-identity=\"urn:uuid:6eff4cba-09bf-4c84-8892-38a06c0c6495\" data-concept-route=\"/concept-identities#concept-6eff4cba-09bf-4c84-8892-38a06c0c6495\">生成元(generator)</u>。如果 $G$ 是一个有限的循环群，$a$ 是一个生成元，那么存在一个正整数 $m$，使得 $a^m=e$。如果 $m$ 是最小的这样的整数，那么对于任意 $g \\in G$，都可以唯一地写成 $g=a^i$，其中 $1 \\leq i \\leq m$，因为如果 $g=a^i=a^j$，且 $1 \\leq i<j \\leq m$，则有矛盾 $a^{j-i}=e$，而 $1 \\leq(j-i)<m$。在这种情况下，群被表示为 $C_m$，其阶数为 $\\left|C_m\\right|=m$。模 $k$ 的加法群是一个阶数为 $k$ 的循环群，但在这种情况下，$a^n$ 的表示方式被替换为\n>\n> $$\n> n a=\\underbrace{a+a+\\cdots+a}_{n} \\bmod k .\n> $$",
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