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aria-label="群"><!--[--><!--[--><!--]--><!--]-->群<!--[--><!--[--><!--]--><!--]--></a><ul style="" class="vp-sidebar-children"><!--[--><li><a class="route-link auto-link vp-sidebar-item" href="#群的定义" aria-label="群的定义"><!--[--><!--[--><!--]--><!--]-->群的定义<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="#阿贝尔群" aria-label="阿贝尔群"><!--[--><!--[--><!--]--><!--]-->阿贝尔群<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="#重复运算的简化" aria-label="重复运算的简化"><!--[--><!--[--><!--]--><!--]-->重复运算的简化<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="#常用群" aria-label="常用群"><!--[--><!--[--><!--]--><!--]-->常用群<!--[--><!--[--><!--]--><!--]--></a><!----></li><!--]--></ul></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/math/number_theory_1.html" aria-label="数论（一）"><!--[--><!--[--><!--]--><!--]-->数论（一）<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/math/number_theory_2.html" aria-label="数论（二）"><!--[--><!--[--><!--]--><!--]-->数论（二）<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/math/number_theory_3.html" aria-label="数论（三）"><!--[--><!--[--><!--]--><!--]-->数论（三）<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/math/probability.html" aria-label="概率"><!--[--><!--[--><!--]--><!--]-->概率<!--[--><!--[--><!--]--><!--]--></a><!----></li><!--]--></ul></li><li><p tabindex="0" class="vp-sidebar-item collapsible">密码学 <span class="right arrow"></span></p><ul style="display:none;" class="vp-sidebar-children"><!--[--><li><a class="route-link auto-link vp-sidebar-item" href="/note/cryptography/rsa.html" aria-label="RSA"><!--[--><!--[--><!--]--><!--]-->RSA<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/cryptography/flip_coin.html" aria-label="抛币协议"><!--[--><!--[--><!--]--><!--]-->抛币协议<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/cryptography/mental_poker.html" aria-label="智能扑克协议"><!--[--><!--[--><!--]--><!--]-->智能扑克协议<!--[--><!--[--><!--]--><!--]--></a><!----></li><!--]--></ul></li><li><p tabindex="0" class="vp-sidebar-item collapsible">图形学 <span class="right arrow"></span></p><ul style="display:none;" class="vp-sidebar-children"><!--[--><li><p tabindex="0" class="vp-sidebar-item collapsible">数学基础 <span class="right arrow"></span></p><ul style="display:none;" class="vp-sidebar-children"><!--[--><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/math/math_01.html" aria-label="矢量"><!--[--><!--[--><!--]--><!--]-->矢量<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/math/math_02.html" aria-label="矩阵"><!--[--><!--[--><!--]--><!--]-->矩阵<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/math/math_03.html" aria-label="立体角"><!--[--><!--[--><!--]--><!--]-->立体角<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/math/transform_01.html" aria-label="几何变换(一)"><!--[--><!--[--><!--]--><!--]-->几何变换(一)<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/math/transform_02.html" aria-label="几何变换(二)"><!--[--><!--[--><!--]--><!--]-->几何变换(二)<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/math/transform_03.html" aria-label="法线变换"><!--[--><!--[--><!--]--><!--]-->法线变换<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/math/transform_04.html" aria-label="摄像机变换"><!--[--><!--[--><!--]--><!--]-->摄像机变换<!--[--><!--[--><!--]--><!--]--></a><!----></li><!--]--></ul></li><li><p tabindex="0" class="vp-sidebar-item collapsible">光照模型 <span class="right arrow"></span></p><ul style="display:none;" class="vp-sidebar-children"><!--[--><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/illumination_model/classic/illumination_model_01.html" aria-label="传统光照模型"><!--[--><!--[--><!--]--><!--]-->传统光照模型<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/illumination_model/luminosity/luminosity.html" aria-label="光度学"><!--[--><!--[--><!--]--><!--]-->光度学<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/illumination_model/BRDF/BRDF.html" aria-label="双向反射分布函数(BRDF)"><!--[--><!--[--><!--]--><!--]-->双向反射分布函数(BRDF)<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/illumination_model/microfacets/microfacets_1.html" aria-label="微平面理论(一)"><!--[--><!--[--><!--]--><!--]-->微平面理论(一)<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/illumination_model/microfacets/microfacets_2.html" aria-label="微平面理论(二)"><!--[--><!--[--><!--]--><!--]-->微平面理论(二)<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/illumination_model/microfacets/microfacets_3.html" aria-label="微平面理论(三)"><!--[--><!--[--><!--]--><!--]-->微平面理论(三)<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/illumination_model/render_function/render_function_1.html" aria-label="光照方程"><!--[--><!--[--><!--]--><!--]-->光照方程<!--[--><!--[--><!--]--><!--]--></a><!----></li><!--]--></ul></li><li><p tabindex="0" class="vp-sidebar-item collapsible">环境光渲染 <span class="right arrow"></span></p><ul style="display:none;" class="vp-sidebar-children"><!--[--><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/image_based_lighting/ibl_01.html" aria-label="环境光渲染(一)"><!--[--><!--[--><!--]--><!--]-->环境光渲染(一)<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/image_based_lighting/ibl_02.html" aria-label="环境光渲染(二)"><!--[--><!--[--><!--]--><!--]-->环境光渲染(二)<!--[--><!--[--><!--]--><!--]--></a><!----></li><!--]--></ul></li><!--]--></ul></li><li><p tabindex="0" class="vp-sidebar-item collapsible">编程语言 <span class="right arrow"></span></p><ul style="display:none;" class="vp-sidebar-children"><!--[--><li><p tabindex="0" class="vp-sidebar-item collapsible">JavaScript <span class="right arrow"></span></p><ul style="display:none;" class="vp-sidebar-children"><!--[--><li><a class="route-link auto-link vp-sidebar-item" href="/note/language/javascript/helloworld.html" aria-label="环境搭建"><!--[--><!--[--><!--]--><!--]-->环境搭建<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/language/javascript/grammar.html" aria-label="基本语法"><!--[--><!--[--><!--]--><!--]-->基本语法<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/language/javascript/function.html" aria-label="函数"><!--[--><!--[--><!--]--><!--]-->函数<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/language/javascript/object_and_class.html" aria-label="对象和类"><!--[--><!--[--><!--]--><!--]-->对象和类<!--[--><!--[--><!--]--><!--]--></a><!----></li><!--]--></ul></li><!--]--></ul></li><!--]--></ul></li><!--]--></ul><!--[--><!--]--></aside><!--]--><!--[--><main class="vp-page"><!--[--><!--]--><div vp-content><!--[--><!--]--><div><h1 id="群-group" tabindex="-1"><a class="header-anchor" href="#群-group"><span>群(Group)</span></a></h1><hr><h2 id="群的定义" tabindex="-1"><a class="header-anchor" href="#群的定义"><span>群的定义</span></a></h2><p>群(<a href="https://en.wikipedia.org/wiki/Group_%28mathematics%29" target="_blank" rel="noopener noreferrer">Group</a>)是一个对象集合G，并且包含了一种运算○，定义如下</p><p>集合<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D43A TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math></mjx-assistive-mml></mjx-container>和运算<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2218"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∘</mo></math></mjx-assistive-mml></mjx-container>构成，一起成为群<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D43A TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="2"><mjx-c class="mjx-c2218"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>G</mi><mo>,</mo><mo>∘</mo><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container>，前提时满足下面的条件</p><ol><li><strong>封闭律</strong>：<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-n"><mjx-c class="mjx-c2200"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2208"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D43A TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">∀</mi><mi>a</mi><mo>,</mo><mi>b</mi><mo>∈</mo><mi>G</mi></math></mjx-assistive-mml></mjx-container>，有<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2218"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="3"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2208"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D43A TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>∘</mo><mi>b</mi><mo>∈</mo><mi>G</mi></math></mjx-assistive-mml></mjx-container></li><li><strong>结合律</strong>：<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-n"><mjx-c class="mjx-c2200"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D450 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2208"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D43A TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">∀</mi><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>∈</mo><mi>G</mi></math></mjx-assistive-mml></mjx-container>，有<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2218"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2218"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="3"><mjx-c class="mjx-c1D450 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2218"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="3"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2218"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="3"><mjx-c class="mjx-c1D450 TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>∘</mo><mo stretchy="false">(</mo><mi>b</mi><mo>∘</mo><mi>c</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>a</mi><mo>∘</mo><mi>b</mi><mo stretchy="false">)</mo><mo>∘</mo><mi>c</mi></math></mjx-assistive-mml></mjx-container></li><li><strong>单位元律</strong>：存在唯一元素<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D452 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2208"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D43A TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mo>∈</mo><mi>G</mi></math></mjx-assistive-mml></mjx-container>，使得<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-n"><mjx-c class="mjx-c2200"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2208"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D43A TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">∀</mi><mi>a</mi><mo>∈</mo><mi>G</mi></math></mjx-assistive-mml></mjx-container>，均有<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2218"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="3"><mjx-c class="mjx-c1D452 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D452 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2218"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="3"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>∘</mo><mi>e</mi><mo>=</mo><mi>e</mi><mo>∘</mo><mi>a</mi><mo>=</mo><mi>a</mi></math></mjx-assistive-mml></mjx-container>，元素<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D452 TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi></math></mjx-assistive-mml></mjx-container>成为单位元</li><li><strong>可逆律</strong>：<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-n"><mjx-c class="mjx-c2200"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2208"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D43A TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">∀</mi><mi>a</mi><mo>∈</mo><mi>G</mi></math></mjx-assistive-mml></mjx-container>，存在<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-n"><mjx-c class="mjx-c2203"></mjx-c></mjx-mi><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.363em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msup><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2208"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D43A TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">∃</mi><msup><mi>a</mi><mrow data-mjx-texclass="ORD"><mo>−</mo><mn>1</mn></mrow></msup><mo>∈</mo><mi>G</mi></math></mjx-assistive-mml></mjx-container>，使得<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2218"></mjx-c></mjx-mo><mjx-msup space="3"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.363em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msup><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-msup space="4"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.363em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msup><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2218"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="3"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D452 TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>∘</mo><msup><mi>a</mi><mrow data-mjx-texclass="ORD"><mo>−</mo><mn>1</mn></mrow></msup><mo>=</mo><msup><mi>a</mi><mrow data-mjx-texclass="ORD"><mo>−</mo><mn>1</mn></mrow></msup><mo>∘</mo><mi>a</mi><mo>=</mo><mi>e</mi></math></mjx-assistive-mml></mjx-container>，其中<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.363em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msup></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>a</mi><mrow data-mjx-texclass="ORD"><mo>−</mo><mn>1</mn></mrow></msup></math></mjx-assistive-mml></mjx-container>称为<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math></mjx-assistive-mml></mjx-container>的逆元</li></ol><p>在表示群<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D43A TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="2"><mjx-c class="mjx-c2218"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>G</mi><mo>,</mo><mo>∘</mo><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container>时，通常省略运算符号，用<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D43A TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math></mjx-assistive-mml></mjx-container>表示</p><h2 id="阿贝尔群" tabindex="-1"><a class="header-anchor" href="#阿贝尔群"><span>阿贝尔群</span></a></h2><p>如果群中的元素满足交换律，也就是对所有的<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2208"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D43A TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>,</mo><mi>b</mi><mo>∈</mo><mi>G</mi></math></mjx-assistive-mml></mjx-container>,均有<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2218"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="3"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2218"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="3"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>∘</mo><mi>b</mi><mo>=</mo><mi>b</mi><mo>∘</mo><mi>a</mi></math></mjx-assistive-mml></mjx-container>,那么该群称为阿贝尔群（<a href="https://en.wikipedia.org/wiki/Abelian_group" target="_blank" rel="noopener noreferrer">Abelian group</a>)</p><h2 id="重复运算的简化" tabindex="-1"><a class="header-anchor" href="#重复运算的简化"><span>重复运算的简化</span></a></h2><p>令<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D43A TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math></mjx-assistive-mml></mjx-container>是运算<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2218"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∘</mo></math></mjx-assistive-mml></mjx-container>下的一个群，那么对于任一元素<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2208"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D43A TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>∈</mo><mi>G</mi></math></mjx-assistive-mml></mjx-container>，非负整数<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2208"></mjx-c></mjx-mo><mjx-TeXAtom space="4" texclass="ORD"><mjx-mi class="mjx-ds mjx-b"><mjx-c class="mjx-c2115 TEX-A"></mjx-c></mjx-mi></mjx-TeXAtom></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mo>∈</mo><mrow data-mjx-texclass="ORD"><mi mathvariant="double-struck">N</mi></mrow></math></mjx-assistive-mml></mjx-container>, a和自己做i次<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2218"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∘</mo></math></mjx-assistive-mml></mjx-container>运算,可以表达为<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.363em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-TeXAtom></mjx-script></mjx-msup><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2208"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D43A TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>a</mi><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msup><mo>∈</mo><mi>G</mi></math></mjx-assistive-mml></mjx-container>, <mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.363em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-TeXAtom></mjx-script></mjx-msup><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2218"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="3"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2218"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="3"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2218"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2026"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2218"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="3"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>a</mi><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msup><mo>=</mo><mi>a</mi><mo>∘</mo><mi>a</mi><mo>∘</mo><mi>a</mi><mo>∘</mo><mo>…</mo><mo>∘</mo><mi>a</mi></math></mjx-assistive-mml></mjx-container>，对于加法群，<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.363em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msup><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-cD7"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="3"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>a</mi><mi>i</mi></msup><mo>=</mo><mi>a</mi><mo>×</mo><mi>i</mi></math></mjx-assistive-mml></mjx-container></p><h2 id="常用群" tabindex="-1"><a class="header-anchor" href="#常用群"><span>常用群</span></a></h2><ol><li><strong>整数群</strong>: 整数集<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-TeXAtom texclass="ORD"><mjx-mi class="mjx-ds mjx-b"><mjx-c class="mjx-c2124 TEX-A"></mjx-c></mjx-mi></mjx-TeXAtom></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mi mathvariant="double-struck">Z</mi></mrow></math></mjx-assistive-mml></mjx-container>在加法下的群<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mi class="mjx-ds mjx-b"><mjx-c class="mjx-c2124 TEX-A"></mjx-c></mjx-mi></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="2"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mi mathvariant="double-struck">Z</mi></mrow><mo>,</mo><mo>+</mo><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container>，其中<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D452 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-msup space="2"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.363em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msup><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mo>=</mo><mn>0</mn><mo>,</mo><msup><mi>a</mi><mrow data-mjx-texclass="ORD"><mo>−</mo><mn>1</mn></mrow></msup><mo>=</mo><mo>−</mo><mi>a</mi></math></mjx-assistive-mml></mjx-container>,同样</li><li><strong>有理数群</strong>: 可以表达为两个整数的商的数称为有理数群，用<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-TeXAtom texclass="ORD"><mjx-mi class="mjx-ds mjx-b"><mjx-c class="mjx-c211A TEX-A"></mjx-c></mjx-mi></mjx-TeXAtom></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mi mathvariant="double-struck">Q</mi></mrow></math></mjx-assistive-mml></mjx-container>表示，其他还有自然数群<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-TeXAtom texclass="ORD"><mjx-mi class="mjx-ds mjx-b"><mjx-c class="mjx-c2115 TEX-A"></mjx-c></mjx-mi></mjx-TeXAtom></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mi mathvariant="double-struck">N</mi></mrow></math></mjx-assistive-mml></mjx-container>, 实数群<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-TeXAtom texclass="ORD"><mjx-mi class="mjx-ds mjx-b"><mjx-c class="mjx-c211D TEX-A"></mjx-c></mjx-mi></mjx-TeXAtom></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mi mathvariant="double-struck">R</mi></mrow></math></mjx-assistive-mml></mjx-container>，复数群<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-TeXAtom texclass="ORD"><mjx-mi class="mjx-ds mjx-b"><mjx-c class="mjx-c2102 TEX-A"></mjx-c></mjx-mi></mjx-TeXAtom></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mi mathvariant="double-struck">C</mi></mrow></math></mjx-assistive-mml></mjx-container>，在加法运算下<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-TeXAtom texclass="ORD"><mjx-mi class="mjx-ds mjx-b"><mjx-c class="mjx-c2124 TEX-A"></mjx-c></mjx-mi></mjx-TeXAtom><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2286"></mjx-c></mjx-mo><mjx-TeXAtom space="4" texclass="ORD"><mjx-mi class="mjx-ds mjx-b"><mjx-c class="mjx-c211A TEX-A"></mjx-c></mjx-mi></mjx-TeXAtom><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2286"></mjx-c></mjx-mo><mjx-TeXAtom space="4" texclass="ORD"><mjx-mi class="mjx-ds mjx-b"><mjx-c class="mjx-c211D TEX-A"></mjx-c></mjx-mi></mjx-TeXAtom><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2286"></mjx-c></mjx-mo><mjx-TeXAtom space="4" texclass="ORD"><mjx-mi class="mjx-ds mjx-b"><mjx-c class="mjx-c2102 TEX-A"></mjx-c></mjx-mi></mjx-TeXAtom></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mi mathvariant="double-struck">Z</mi></mrow><mo>⊆</mo><mrow data-mjx-texclass="ORD"><mi mathvariant="double-struck">Q</mi></mrow><mo>⊆</mo><mrow data-mjx-texclass="ORD"><mi mathvariant="double-struck">R</mi></mrow><mo>⊆</mo><mrow data-mjx-texclass="ORD"><mi mathvariant="double-struck">C</mi></mrow></math></mjx-assistive-mml></mjx-container></li><li><strong>整数模n加法群</strong>: 对于任意<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2265"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>≥</mo><mn>1</mn></math></mjx-assistive-mml></mjx-container>，<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-TeXAtom texclass="ORD"><mjx-mi class="mjx-ds mjx-b"><mjx-c class="mjx-c2124 TEX-A"></mjx-c></mjx-mi></mjx-TeXAtom><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow data-mjx-texclass="ORD"><mi mathvariant="double-struck">Z</mi></mrow><mi>n</mi></msub></math></mjx-assistive-mml></mjx-container>表示所有整数模n的集合，完整表示为<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-TeXAtom texclass="ORD"><mjx-mi class="mjx-ds mjx-b"><mjx-c class="mjx-c2124 TEX-A"></mjx-c></mjx-mi></mjx-TeXAtom><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="2"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mspace style="width:0.667em;"></mjx-mspace><mjx-mi class="mjx-n"><mjx-c class="mjx-c6D"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c64"></mjx-c></mjx-mi><mjx-mstyle><mjx-mspace style="width:0.167em;"></mjx-mspace></mjx-mstyle><mjx-mstyle><mjx-mspace style="width:0.167em;"></mjx-mspace></mjx-mstyle><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><msub><mrow data-mjx-texclass="ORD"><mi mathvariant="double-struck">Z</mi></mrow><mi>n</mi></msub><mo>,</mo><mo>+</mo><mo stretchy="false">(</mo><mspace width="0.667em"></mspace><mi>mod</mi><mstyle scriptlevel="0"><mspace width="0.167em"></mspace></mstyle><mstyle scriptlevel="0"><mspace width="0.167em"></mspace></mstyle><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container>，单位元<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D452 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c30"></mjx-c></mjx-mn></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mo>=</mo><mn>0</mn></math></mjx-assistive-mml></mjx-container>，逆元<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.363em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msup><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="3"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>a</mi><mrow data-mjx-texclass="ORD"><mo>−</mo><mn>1</mn></mrow></msup><mo>=</mo><mi>n</mi><mo>−</mo><mi>a</mi></math></mjx-assistive-mml></mjx-container><ol><li><mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-TeXAtom texclass="ORD"><mjx-mi class="mjx-ds mjx-b"><mjx-c class="mjx-c2124 TEX-A"></mjx-c></mjx-mi></mjx-TeXAtom><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c7B"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="2"><mjx-c class="mjx-c2026"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="2"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="3"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c7D"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="2"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mspace style="width:0.667em;"></mjx-mspace><mjx-mi class="mjx-n"><mjx-c class="mjx-c6D"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c64"></mjx-c></mjx-mi><mjx-mstyle><mjx-mspace style="width:0.167em;"></mjx-mspace></mjx-mstyle><mjx-mstyle><mjx-mspace style="width:0.167em;"></mjx-mspace></mjx-mstyle><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow data-mjx-texclass="ORD"><mi mathvariant="double-struck">Z</mi></mrow><mi>n</mi></msub><mo>=</mo><mo stretchy="false">(</mo><mo fence="false" stretchy="false">{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo fence="false" stretchy="false">}</mo><mo>,</mo><mo>+</mo><mo stretchy="false">(</mo><mspace width="0.667em"></mspace><mi>mod</mi><mstyle scriptlevel="0"><mspace width="0.167em"></mspace></mstyle><mstyle scriptlevel="0"><mspace width="0.167em"></mspace></mstyle><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container></li><li><mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-n"><mjx-c class="mjx-c23"></mjx-c></mjx-mi><mjx-msub><mjx-TeXAtom texclass="ORD"><mjx-mi class="mjx-ds mjx-b"><mjx-c class="mjx-c2124 TEX-A"></mjx-c></mjx-mi></mjx-TeXAtom><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">#</mi><msub><mrow data-mjx-texclass="ORD"><mi mathvariant="double-struck">Z</mi></mrow><mi>n</mi></msub><mo>=</mo><mi>n</mi></math></mjx-assistive-mml></mjx-container></li></ol></li><li><strong>整数模n乘法群</strong>(<a href="https://en.wikipedia.org/wiki/Multiplicative_group_of_integers_modulo_n" target="_blank" rel="noopener noreferrer">Multiplicative group of integers modulo n</a>) : <mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-TeXAtom texclass="ORD"><mjx-mi class="mjx-ds mjx-b"><mjx-c class="mjx-c2124 TEX-A"></mjx-c></mjx-mi></mjx-TeXAtom><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow data-mjx-texclass="ORD"><mi mathvariant="double-struck">Z</mi></mrow><mi>n</mi></msub></math></mjx-assistive-mml></mjx-container>中所有与n互质的元素构成一个有限乘法群，这里的乘法指模n乘法，用<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msubsup><mjx-TeXAtom texclass="ORD"><mjx-mi class="mjx-ds mjx-b"><mjx-c class="mjx-c2124 TEX-A"></mjx-c></mjx-mi></mjx-TeXAtom><mjx-script style="vertical-align:-0.247em;margin-left:0;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2217"></mjx-c></mjx-mo></mjx-TeXAtom><mjx-spacer style="margin-top:0.322em;"></mjx-spacer><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msubsup></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow data-mjx-texclass="ORD"><mi mathvariant="double-struck">Z</mi></mrow><mi>n</mi><mrow data-mjx-texclass="ORD"><mo>∗</mo></mrow></msubsup></math></mjx-assistive-mml></mjx-container>表示，例如<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msubsup><mjx-TeXAtom texclass="ORD"><mjx-mi class="mjx-ds mjx-b"><mjx-c class="mjx-c2124 TEX-A"></mjx-c></mjx-mi></mjx-TeXAtom><mjx-script style="vertical-align:-0.263em;margin-left:0;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2217"></mjx-c></mjx-mo></mjx-TeXAtom><mjx-spacer style="margin-top:0.18em;"></mjx-spacer><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c35"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msubsup><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c7B"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c34"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c37"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c38"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c33"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c34"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c7D"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="2"><mjx-c class="mjx-c2217"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mspace style="width:0.667em;"></mjx-mspace><mjx-mi class="mjx-n"><mjx-c class="mjx-c6D"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c64"></mjx-c></mjx-mi><mjx-mstyle><mjx-mspace style="width:0.167em;"></mjx-mspace></mjx-mstyle><mjx-mstyle><mjx-mspace style="width:0.167em;"></mjx-mspace></mjx-mstyle><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c35"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow data-mjx-texclass="ORD"><mi mathvariant="double-struck">Z</mi></mrow><mrow data-mjx-texclass="ORD"><mn>15</mn></mrow><mrow data-mjx-texclass="ORD"><mo>∗</mo></mrow></msubsup><mo>=</mo><mo stretchy="false">(</mo><mo fence="false" stretchy="false">{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>4</mn><mo>,</mo><mn>7</mn><mo>,</mo><mn>8</mn><mo>,</mo><mn>11</mn><mo>,</mo><mn>13</mn><mo>,</mo><mn>14</mn><mo fence="false" stretchy="false">}</mo><mo>,</mo><mo>∗</mo><mo stretchy="false">(</mo><mspace width="0.667em"></mspace><mi>mod</mi><mstyle scriptlevel="0"><mspace width="0.167em"></mspace></mstyle><mstyle scriptlevel="0"><mspace width="0.167em"></mspace></mstyle><mn>15</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container><ol><li>如果p是质数，那么<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msubsup><mjx-TeXAtom texclass="ORD"><mjx-mi class="mjx-ds mjx-b"><mjx-c class="mjx-c2124 TEX-A"></mjx-c></mjx-mi></mjx-TeXAtom><mjx-script style="vertical-align:-0.247em;margin-left:0;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2217"></mjx-c></mjx-mo></mjx-TeXAtom><mjx-spacer style="margin-top:0.322em;"></mjx-spacer><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msubsup><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c7B"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c33"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="2"><mjx-c class="mjx-c2026"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="2"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="3"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c7D"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="2"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mspace style="width:0.667em;"></mjx-mspace><mjx-mi class="mjx-n"><mjx-c class="mjx-c6D"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c64"></mjx-c></mjx-mi><mjx-mstyle><mjx-mspace style="width:0.167em;"></mjx-mspace></mjx-mstyle><mjx-mstyle><mjx-mspace style="width:0.167em;"></mjx-mspace></mjx-mstyle><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mi class="mjx-n" space="2"><mjx-c class="mjx-c23"></mjx-c></mjx-mi><mjx-msubsup><mjx-TeXAtom texclass="ORD"><mjx-mi class="mjx-ds mjx-b"><mjx-c class="mjx-c2124 TEX-A"></mjx-c></mjx-mi></mjx-TeXAtom><mjx-script style="vertical-align:-0.247em;margin-left:0;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2217"></mjx-c></mjx-mo></mjx-TeXAtom><mjx-spacer style="margin-top:0.322em;"></mjx-spacer><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msubsup><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="3"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow data-mjx-texclass="ORD"><mi mathvariant="double-struck">Z</mi></mrow><mi>p</mi><mrow data-mjx-texclass="ORD"><mo>∗</mo></mrow></msubsup><mo>=</mo><mo stretchy="false">(</mo><mo fence="false" stretchy="false">{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>p</mi><mo>−</mo><mn>1</mn><mo fence="false" stretchy="false">}</mo><mo>,</mo><mo>+</mo><mo stretchy="false">(</mo><mspace width="0.667em"></mspace><mi>mod</mi><mstyle scriptlevel="0"><mspace width="0.167em"></mspace></mstyle><mstyle scriptlevel="0"><mspace width="0.167em"></mspace></mstyle><mi>p</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><mi mathvariant="normal">#</mi><msubsup><mrow data-mjx-texclass="ORD"><mi mathvariant="double-struck">Z</mi></mrow><mi>p</mi><mrow data-mjx-texclass="ORD"><mo>∗</mo></mrow></msubsup><mo>=</mo><mi>p</mi><mo>−</mo><mn>1</mn></math></mjx-assistive-mml></mjx-container></li><li>如果<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45E TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mi>q</mi></math></mjx-assistive-mml></mjx-container>是质数，<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45E TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mi>p</mi><mi>q</mi></math></mjx-assistive-mml></mjx-container>，那么根据欧拉定理<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msubsup><mjx-TeXAtom texclass="ORD"><mjx-mi class="mjx-ds mjx-b"><mjx-c class="mjx-c2124 TEX-A"></mjx-c></mjx-mi></mjx-TeXAtom><mjx-script style="vertical-align:-0.247em;margin-left:0;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2217"></mjx-c></mjx-mo></mjx-TeXAtom><mjx-spacer style="margin-top:0.322em;"></mjx-spacer><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msubsup></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow data-mjx-texclass="ORD"><mi mathvariant="double-struck">Z</mi></mrow><mi>n</mi><mrow data-mjx-texclass="ORD"><mo>∗</mo></mrow></msubsup></math></mjx-assistive-mml></mjx-container>里元素的个数为<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D719 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="3"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45E TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="3"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ϕ</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>p</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>q</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container></li><li><mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msubsup><mjx-TeXAtom texclass="ORD"><mjx-mi class="mjx-ds mjx-b"><mjx-c class="mjx-c2124 TEX-A"></mjx-c></mjx-mi></mjx-TeXAtom><mjx-script style="vertical-align:-0.263em;margin-left:0;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2217"></mjx-c></mjx-mo></mjx-TeXAtom><mjx-spacer style="margin-top:0.18em;"></mjx-spacer><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msubsup><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c7B"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c35"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c37"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c7D"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow data-mjx-texclass="ORD"><mi mathvariant="double-struck">Z</mi></mrow><mrow data-mjx-texclass="ORD"><mn>12</mn></mrow><mrow data-mjx-texclass="ORD"><mo>∗</mo></mrow></msubsup><mo>=</mo><mo fence="false" stretchy="false">{</mo><mn>1</mn><mo>,</mo><mn>5</mn><mo>,</mo><mn>7</mn><mo>,</mo><mn>11</mn><mo fence="false" stretchy="false">}</mo></math></mjx-assistive-mml></mjx-container></li></ol></li></ol></div><!--[--><!--]--></div><footer class="vp-page-meta"><!----><div class="vp-meta-item git-info"><div class="vp-meta-item last-updated"><span class="meta-item-label">Last Updated: </span><!----></div><div class="vp-meta-item contributors"><span class="meta-item-label">Contributors: </span><span class="meta-item-info"><!--[--><!--[--><span class="contributor" title="email: thejinchao@gmail.com">thejinchao</span><!----><!--]--><!--]--></span></div></div></footer><nav class="vp-page-nav" aria-label="page navigation"><a class="route-link auto-link prev" href="/note/math/symbol.html" aria-label="常用数学符号"><!--[--><div class="hint"><span class="arrow left"></span> Prev</div><div class="link"><span>常用数学符号</span></div><!--]--></a><a class="route-link auto-link next" 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