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

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. That may seem tedious to you, but think of the extra work it represents for TeX.SX users willing to give you a hand. Help them help you: remove that one hurdle between you and a solution to your problem. – Adam Liter Apr 16 '15 at 3:29A
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