$G \circlearrowleft X$ (or $G \circlearrowright X$) to denote that $G$ acts on $X$.
Edit by A.H. :
Here are some latex definitions that produce the symbol that David Speyer describes in his comment:
\def\acts{
\hspace{.1cm}
{
\setlength{\unitlength}{.30mm}
\linethickness{.09mm}
\begin{picture}(8,8)(0,0)
\qbezier(7,6)(4.5,8.3)(2,7)
\qbezier(2,7)(-1.5,4)(2,1)
\qbezier(2,1)(4.5,-.3)(7,2)
\qbezier(7,6)(6.1,7.5)(6.8,9)
\qbezier(7,6)(5,6.1)(4.2,4.4)
\end{picture}
\hspace{.1cm}
}}
and
\def\acted{
\hspace{.1cm}
{
\setlength{\unitlength}{.30mm}
\linethickness{.09mm}
\begin{picture}(8,8)(0,0)
\qbezier(1,6)(3.5,8.3)(6,7)
\qbezier(6,7)(9.5,4)(6,1)
\qbezier(6,1)(3.5,-.3)(1,2)
\qbezier(1,6)(1.9,7.5)(1.2,9)
\qbezier(1,6)(3,6.1)(3.8,4.4)
\end{picture}
\hspace{.1cm}
}}
