a commutative diagram compatible with group actions

Question

I'm trying to draw the following commutative diagram using xymatrix:

\documentclass{article}
\usepackage[all]{xy}     

\begin{document}

$\xymatrix{A \ar[rr] & & B  \\ & C \ar[ul] \ar[ur] & }$

\end{document}

How can I draw symbols of group action?

Answer 1

Or like this:

% arara: pdflatex

\documentclass{article} 
\usepackage{tikz-cd} 

\begin{document}  
\[
    \begin{tikzcd}  
        A \arrow{rr}\arrow[loop left, "G_1"] & & B \arrow[loop right, "G_2"] \\ 
        & C \arrow{ul} \arrow{ur} \arrow[loop below, "G_3"] & 
    \end{tikzcd}
\] 
\end{document}

Answer 2

Here's a solution using TiKZ instead of xypics. The arrows are drawn as circular arcs with modifiable parameters; the location and style for the labels of the group actions can also be varied.

Some particular remarks on the code:

• The group actions are denoted by nearly complete circles (omitting 45° of the turn), whose orientations can be modified by changing the starting and ending angles.

• The end tips of each arrow are elongated, in order to better match the appearance of the orientation of the arrow-head with the angle at which the arc ends.

Source code

\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{calc}

\begin{document}
\begin{tikzpicture}[-stealth]
  % Position nodes symmetrically about a central point
  \node (A) at ({cos(150)},{sin(150)}) {$A$};
  \node (B) at ({cos(30)},{sin(30)}) {$B$};
  \node (C) at ({cos(270)},{sin(270)}) {$C$};

  % Draw arrows between nodes
  \draw (A) to (B); \draw (C) to (A); \draw (C) to (B);

  % Define co-ordinates for the arrows of the group actions
  \coordinate (G1') at ($(A.west) + (-0.5em,0)$);
  \coordinate (G2') at ($(B.east) + (0.5em,0)$);
  \coordinate (G3') at ($(C.south west) + (0.5em,-2ex)$);

  % Draw circular arcs of 1ex radius for the group actions
  % (tips are slightly lengthened to optimise orientation of arrow head)
  \draw [->, shorten >=-1.5pt] (G1') arc (245:-70:1ex);
  \draw [->, shorten >=-1.5pt] (G2') arc (-65:250:1ex);
  \draw [->, shorten >=-1.5pt] (G3') arc (315:0:1ex);

  % Draw labels for the group actions, smaller than the main nodes
  \node (G1) at ($(G1') + (-1em,0)$) {\footnotesize $G_1$};
  \node (G2) at ($(G2') + (1em,0)$) {\footnotesize $G_2$};
  \node (G3) at ($(G3') + (0,-2ex)$) {\footnotesize $G_3$};
\end{tikzpicture}
\end{document}

Result

Answer 3

Something like this? (With help of this documentation, p. 4, since I'm not very proficient in xy-pic.)

\documentclass{article} 
\usepackage[all,cmtip]{xy} 
\begin{document}  
  \[
    \xymatrix{  A \ar@(dl,ul)[]^{G_1} \ar[rr] & & B \ar@(dr,ur)[]_{G_2} \\ 
      & C \ar@(dl,dr)[]_{G_3}\ar[ul] \ar[ur] & }
  \] 
\end{document}

Edit: following LaRiFari's suggestion, now the cmtip option is used.The arrow heads are much better looking!