$S_3$的子群图

Question

我正在尝试绘制对称群 $S_3$ 的子群图。分析了一些材料，我得到了一些接近的结果，但还远远不够。

下面我介绍我正在使用的命令：

\documentclass[12pt]{article}
\usepackage{tikz}
\usepackage{amsfonts}
\title{Subgroup Diagram of $S_3$}

\begin{document}
\begin{figure}
\centering
\begin{tikzpicture}
 \begin{scope}[rotate=45,scale=1.5,transform shape,nodes={fill=white,transform
   shape=false}]
   \draw (0,0) grid (1,1);
   \path (1,1) node (A1)  {$S_3$}
   (0,2) node (A2)  {$\langle \alpha \rangle$}
   (-1,0) node (A3)  {$\langle \alpha^2 \beta \rangle$}
   (1,0) node (A4)  {$\langle \alpha \beta \rangle$}
   (0,1) node (A5)  {$\langle \beta \rangle$}
   (0,0) node (A6)  {$\langle \{e\} \rangle$};
 \end{scope}
\end{tikzpicture}
\caption{Subgroup Diagram of $S_3$}
\end{figure}
\end{document}

更准确地说，我想要制作的图表如下：

提前非常感谢您。

Answer 1

像这样？

\documentclass[12pt]{article}
\usepackage{tikz}
\usetikzlibrary{arrows.meta,
                calc, chains
                }

\begin{document}
    \begin{figure}
    \centering
\begin{tikzpicture}[
node distance = 12mm and 8mm,
  start chain = A going right,
     N/.style = {circle, inner sep=1pt, on chain=A}
                    ]
\node[N]    {$\langle\alpha\rangle$};
\node[N]    {$\langle\beta\rangle$};
\node[N]    {$\langle\alpha\beta\rangle$};
\node[N]    {$\langle\alpha^2\beta\rangle$};
% node above
\node (a) [above=of {$(A-1.west)!0.5!(A-4)$}]   {$S_3$};
\node (b) [below=of {$(A-1.west)!0.5!(A-4)$}]   {$\{e\}$};
% paths
\foreach    \i in {1,2,3,4}
\draw   (a) -- (A-\i) --  (b);
\end{tikzpicture}
    \caption{Subgroup Diagram of $S_3$}
    \end{figure}
\end{document}

Answer 2

如果改变子组的顺序，你会得到更对称的图表：

\documentclass[12pt]{article}
\usepackage{tikz-cd}

\begin{document}

\begin{figure}
\centering

\begin{tikzcd}[
  row sep=1em,
  column sep={4em,between origins},
  arrows=no head,
]
&&[-1em] S_3 \arrow[lld] \arrow[ld] \arrow[rd] \arrow[rrd] &[-1em] \\
\langle \alpha \rangle &
\langle \alpha\beta \rangle &&
\langle \alpha^2\beta \rangle &
\langle \beta \rangle \\
&& \langle e\rangle \arrow[llu] \arrow[lu] \arrow[ru] \arrow[rru]  
\end{tikzcd}

\caption{Subgroup Diagram of $S_3$}

\end{figure}

\end{document}

Answer 3

或者使用tikz-cd：

\documentclass[12pt]{article}
\usepackage{tikz-cd}

\title{Subgroup Diagram of $S_3$}
\begin{document}
\begin{figure}
\centering
\begin{tikzcd}[row sep= 1em,column sep=1em]
  &   & S_3 \arrow[lld, no head] \arrow[ld, no head] \arrow[d, no head] \arrow[rd, no head] &   \\
\langle \alpha \rangle & \langle \beta \rangle & \langle \alpha \beta \rangle                                                                                 & \langle \alpha^2 \beta \rangle \\
  &   & \langle \{e\} \rangle\arrow[llu, no head] \arrow[lu, no head] \arrow[u, no head] \arrow[ru, no head] &  
\end{tikzcd}
\caption{Subgroup Diagram of $S_3$}
\end{figure}
\end{document}

附录 2022/5/24：这里有使用tikz-node的对称版本。

\documentclass[a4paper,12pt]{article}
\usepackage{tikz}

\title{Subgroup Diagram of $S_3$}
\begin{document}
\begin{figure}
\centering
\begin{tikzpicture}[scale=.7]
  \node (one) at (0,2) {$S_3$};
  \node (a) at (-3,0) {$\langle \alpha \rangle$};
  \node (b) at (-1,0) {$\langle \beta \rangle$};
  \node (c) at (1,0) {$\langle \alpha \beta \rangle$};
  \node (d) at (3,0) {$\langle \alpha^2 \beta \rangle$};
  \node (zero) at (0,-2) {$\langle \{e\} \rangle$};
  \draw (zero) -- (a) -- (one) -- (b) -- (zero) -- (c) -- (one) -- (d) -- (zero);
\end{tikzpicture}
\caption{Subgroup Diagram of $S_3$}
\end{figure}
\end{document}

Answer 4

TikZgraphs和graphdrawing库也非常适合此目的（另请参阅“第四部分图形绘制”手册)。需要注意的是，需要使用 LuaLaTeX graphdrawing，因为它的布局算法是用 lua 编写的。

\documentclass[tikz]{standalone}

\usetikzlibrary{graphs,graphdrawing}
\usegdlibrary{layered}

% Load this library before loading polyglossia (or babel), if you're using one of those.
% This is required to fix their messing with the double quote (") character for shortcuts.
% \usetikzlibrary{babel}

\begin{document}
\begin{tikzpicture}
    % \begin{noindent} % so that latexindent doesn't mess this up
    \graph [layered layout] {
        "\(S_3\)"
        --
        {
            "\(\langle \alpha \rangle\)",
            "\(\langle \beta \rangle\)",
            "\(\langle \alpha\beta \rangle\)",
            "\(\langle \alpha^2\beta \rangle\)",
        }
        --
        "\(\{e\}\)";
    };
    % \end{noindent}
\end{tikzpicture}
\end{document}