Dummit and Foote clearly define it using the operation of that same group. What book are you using?
Are you suggesting that your book defines a group action of (G,\ast_1) on X via a function \phi : G \times X \to X such that \phi(g,\phi(h,x))=\phi(g\ast_2h,x), where g,h\in G; x\in X, and...