group action

A group action is a way to describe how a mathematical group—an organized set of elements with a structured way to combine them—"acts" on a set of objects, transforming or moving them while respecting the group's rules. Think of it like a team of players (group elements) performing moves on a game board (the set), where each move follows certain rules, and applying different moves changes the board's state while preserving the overall structure. Group actions help us understand symmetry, patterns, and the way objects can be transformed consistently within mathematical systems.