Subgroup diagram of $S_3$

Question

I am trying to make the subgroup diagram of the symmetric group $S_3$. Analyzing some materials, I arrived at something close, but it is still far from desired.

Below I present the commands that I am using:

\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}

More precisely the diagram I would like to make is the following:

Thank you very much in advance.

Answer 1

Like this?

\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

You get a more symmetric diagram if you change the order of the subgroups:

\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

Or using 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}

Addendum 2022/5/24: Here there is the symmetrical version using tikz-node.

\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

The TikZ graphs and graphdrawing libraries are nice for this as well (see also "Part IV Graph Drawing" of the pgfmanual). One caveat is the requirement of LuaLaTeX for graphdrawing, since its layouting algorithms are written in lua.

\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}