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我怎样才能用 LaTeX 来表达这个公式？

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我怎样才能用 LaTeX 来表达这个公式？

同构

我不确定如何在 LaTeX 中用注释来标注同构。这是我在 Latex 中显示同构部分的方式。但我不知道如何向其中添加绿色内容。

\documentclass{article}
\usepackage{amssymb}
\usepackage{tikz}
\newcommand{\quo}[2]{{\raisebox{.2em}{$#1$}\left/\raisebox{-.2em}{$#2$}\right.}} % The quotient of elements or groups
\begin{document}
$\quo{HK}{K}\simeq\quo{H}{H\cap K}$\\
\end{document}

答案1

例如，看看这个：

\documentclass{article}
\usepackage{amsmath,amssymb}
\usepackage{mathtools}

\newcommand{\quo}[2]{{\raisebox{.2em}{$#1\!$}\left/\raisebox{-.2em}{$#2$}\right.}} % The quotient of elements or groups

\usepackage{tikz}
\usetikzlibrary{decorations.pathreplacing}
\usetikzlibrary{calc,tikzmark}
\tikzset{%
    pics/mybraces/.style args={#1/#2}{code={%
            \draw [decorate,decoration={brace,amplitude=4pt,#2},rotate=#1] (0,0) -- (0,1);
    }},
    pics/mybraces/.default=30/{}
} 

\begin{document}
    You can do whatever you want with \LaTeX{}:
    \[
    \underbrace{\quo{\overbrace{HK}^{G}}{K}}_{\mathllap{\mathrm{PGL}(2,\mathbb{C})\text{: projective linear group}}} \simeq \overbrace{\quo{H}{H\cap K}}^{\mathrlap{\mathrm{PSL}(2,\mathbb{C})\text{: projective special linear group}}}\mkern-58mu \underbrace{\phantom{\raisebox{-.2em}{$H\cap K$}}}_{\{\pm I_2\}}
    \]
    You can use also \textbackslash{}\texttt{tikzmark}:
    \[
    \tikzmark{leftpoint}\quo{\overbrace{HK}^{G}}{K} \simeq \quo{H}{H\cap K}\tikzmark{rightpoint}\mkern-58mu \underbrace{\phantom{\raisebox{-.2em}{$H\cap K$}}}_{\{\pm I_2\}}
    \]
    \begin{tikzpicture}[overlay,remember picture]
        \pic at ($(pic cs:leftpoint)+(4pt,-2.5ex)$) {mybraces};
        \node[text width=100pt, align=right] at ($(pic cs:leftpoint)+(-60pt,-1.5ex)$) {$_{\mathrm{PGL}(2,\mathbb{C})\text{:}}$\\ \scriptsize projective linear group};
        \pic at ($(pic cs:rightpoint)+(7pt,-1ex)$) {mybraces={30/mirror}};
        \node[text width=130pt, align=left] at ($(pic cs:rightpoint)+(75pt,3ex)$) {$_{\mathrm{PSL}(2,\mathbb{C})\text{:}}$\\ \scriptsize projective special linear group};
    \end{tikzpicture}%
    However, as Andrew Swan \& Co. already told you, in your case something like this is better:
    \begin{align*} 
        \quo{HK}{K}&\simeq \quo{H}{H\cap K}
        \intertext{where:}
        HK &= G \\
        \intertext{the projective linear group is represented by:} 
        \quo{HK}{K}&= \mathrm{PGL}(2,\mathbb{C}) 
        \intertext{the projective special linear group is represented by:} 
        \quo{H}{H\cap K}&= \mathrm{PSL}(2,\mathbb{C}) 
        \intertext{and} 
        H\cap K&=\{\pm I_2\}
    \end{align*}
\end{document}

在此处输入图片描述

答案2

正如 Andrew Swann 的评论，您应该用一些文字来解释该公式。无论如何，我认为您可以使用这个：

\begin{equation}
    \underbrace{HK}_{G}/\underbrace{K}_{\mathrm{PGL}(2,\mathbb{C})} \simeq \underbrace{H}_{PSL(2,\mathbb{C})}/\underbrace{H\cap K}_{\pm I_2}
\end{equation}

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