Group actions generalize group multiplication. If \(G\) is a group and \(X\) is an arbitrary set, a group action of an element \(g \in G\) and \(x \in X\) is a product, \(gx\text{,}\) living in \(X\text{.}\) Many problems in algebra are best be attacked via group actions. For example, the proofs of the Sylow theorems and of Burnside’s Counting Theorem are most easily understood when they are formulated in terms of group actions.
