Group actions

A group action is a way to describe how a set of symmetries or transformations (called a group) can be applied to a set of objects in a consistent manner. Think of it as a formal way to understand how elements like rotations or flips can move or change objects without destroying their structure. For example, the symmetries of a square—rotations and reflections—can be represented as a group that acts on the square’s corners, revealing patterns and invariants. Essentially, group actions connect the abstract idea of symmetry with concrete changes to objects.