group action
Think of a group as a collection of symmetries — rotations of a cube, reshufflings of a deck of cards — and think of a set as the things those symmetries move around. A group action is the formal way of saying “this group rearranges that set.” Each group element becomes a way to permute the set, and composing group elements composes the permutations, so the group structure is faithfully reflected in how the set gets shuffled.
Precisely, an action of a group G on a set X is a homomorphism from G to the symmetric group Sym(X) of all bijections of X. Equivalently it is a map G x X -> X, written (g, x) -> g·x, satisfying e·x = x for the identity e and g·(h·x) = (gh)·x for all g, h in G. The second condition is exactly what makes the assignment g -> (x -> g·x) a homomorphism rather than just a map.
Actions are the bridge between abstract groups and concrete combinatorics or geometry. The two basic invariants of an action — its orbits (which points can be reached from which) and its stabilizers (which elements fix a point) — control almost everything, and the orbit-stabilizer theorem ties them together. An action is called faithful when the homomorphism G -> Sym(X) is injective, so distinct group elements really do act differently; many structure theorems begin by producing a faithful action.
A caveat: the same group can act on the same set in genuinely different ways, and these are different actions, not different descriptions of one thing. For example Z/4Z can act on four points as a single 4-cycle, or trivially (every element fixes every point); both satisfy the axioms. The action, not just the group, is the object of study.
S_3 acts on the set {1, 2, 3} by permuting the labels: the transposition (1 2) sends 1 to 2, 2 to 1, and fixes 3. This action is faithful, since the only permutation fixing all three points is the identity, so the homomorphism S_3 -> Sym({1,2,3}) is an isomorphism here.
The defining action of S_3 on three letters.
Left and right actions differ in the composition rule: a left action satisfies g·(h·x) = (gh)·x, while a right action satisfies (x·g)·h = x·(gh). Any right action becomes a left action via g·x := x·g^{-1}, so the choice is largely a convention, but mixing them silently causes sign-of-composition errors.