3

the isomorphism

I'm not sure how to text the isomorphism with notes in LaTeX. This is how I show the isomorphism part in Latex. But I don't know how to add the green content to it.

\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}
12
  • 1
    Welcome to TeX.SE. Could you please provide a MWE (Minimal Working Example) of what you have tried already.
    – Jan
    Jan 24, 2017 at 7:01
  • I am not a mathematician, so I may be wrong. In my eyes, this looks like the green content was added as explanation of the blue formulae. Do you want to reassemble this look in LaTeX?
    – Jan
    Jan 24, 2017 at 7:03
  • Hello, guys. I recently learn Latex by myself, but I got some problems in texting some of my notes in Latex. So, I just sign up this website asking for help. This is one of the part of my note. What I want to do is to reassemble the part "after" the comma, including the green content, but I don't know how can I do this.
    – Coldjoke
    Jan 24, 2017 at 7:11
  • I can already do the part of the isomorphism and the quotient group by applying a article in the website.
    – Coldjoke
    Jan 24, 2017 at 7:13
  • 1
    I don't think you should be aiming to reproduce the hand-written layout. Use extra sentences/clauses to add explanation before/after the equation. Jan 24, 2017 at 7:36

2 Answers 2

4

Look at this, for example:

\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}

enter image description here

5
  • Thank you for helping me solving the problem! :) I'm wondering if the '\overbrace' and '\underbrace' can be replaced by a 90-degree-rotated-bracket?
    – Coldjoke
    Jan 24, 2017 at 12:14
  • btw PSL is short for projective special linear group. :)
    – Coldjoke
    Jan 24, 2017 at 12:15
  • @Coldjoke I've redone the first example and added one with tikz, you can rotate the bracket as you like!
    – CarLaTeX
    Jan 24, 2017 at 21:39
  • Thanks a lot for your help. I'll go study more about the package 'tikz' :)
    – Coldjoke
    Jan 25, 2017 at 14:42
  • @Coldjoke You're welcome! TikZ is a very powerful package, you can do a lot of extraordinary thing with it!
    – CarLaTeX
    Jan 25, 2017 at 14:46
0

As Andrew Swann's comment, you should explain the formula by some words. Anyway, I think you can use this:

\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}
2
  • Wow! I don't even think of this before. I can try this with some explanation Andrew Swann and Jan recommend. Thanks all of you for helping me:)
    – Coldjoke
    Jan 24, 2017 at 7:55
  • @CarLaTeX OK! Got it!
    – Coldjoke
    Jan 24, 2017 at 8:05

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