Turn abstract multiplication into motion of a set

Try

The symmetries of a square act on its four vertices. A chosen vertex can move to any of the four positions, but some symmetries leave it fixed. These two observations measure different things: the orbit records reachable positions, while the stabilizer records transformations that leave the chosen position unchanged.

There are eight symmetries, four possible positions, and two symmetries fixing one vertex. Predict why these counts multiply rather than add.

Predict

If two group elements move x to the same point, what does their relative product do to x? For a left action, g·x=h·x implies h⁻¹g fixes x. This suggests grouping transformations by left cosets of the stabilizer. The stabilizer need not be normal, because we need a coset set rather than a quotient group.

Experiment

For D₄, label vertices modulo four and use r(i)=i+1,s(i)=−i. The stabilizer of zero is ｛e,s｝.

Coset of stabilizer	Image of vertex zero
｛e,s｝	0
｛r,rs｝	1
｛r²,r²s｝	2
｛r³,r³s｝	3

Each pair consists of exactly the transformations giving the same image. Check a representative change explicitly: r and rs both send zero to one. This is the mechanism of the orbit-stabilizer theorem, not a coincidence of the square.

Understand

A left action of G on X assigns g·x∈X with e·x=x and (gh)·x=g·(h·x). Each g acts bijectively, with inverse action by g⁻¹. The orbit G·x is the set of all g·x. The stabilizer G_x is the subgroup of elements fixing x: identity fixes x, and if a,b fix it, then ab⁻¹ does too.

Define gG_x↦g·x. Equal cosets imply h⁻¹g∈G_x, hence equal images. Conversely equal images imply that same membership and equal cosets. The map is onto the orbit by its definition. Thus the orbit is in bijection with the left cosets of the stabilizer. For finite G, |G·x|=|G|/|G_x|.

An action is transitive when X is one orbit. It is faithful when only identity fixes every point, equivalently the associated map G→Sym(X) has trivial kernel. These properties are independent: an action on a singleton is transitive but is not faithful for a nontrivial group. The natural action of Sₙ on its n letters is both faithful and transitive when n≥1.

Every subgroup H gives an action on left cosets by g·aH=gaH. Representative independence follows because aH=bH implies gaH=gbH. An element g fixes every coset exactly when a⁻¹ga∈H for every a. Therefore the action kernel is the intersection of all aHa⁻¹, called the core of H. It is the largest normal subgroup of G contained in H.

Apply

Choose a different square vertex and compute its stabilizer. Compare it with the first stabilizer by conjugation. In a transitive action, stabilizers of different points are conjugate, not necessarily equal. Prove this by checking which transformations fix g·x.

When an orbit count looks wrong, audit whether you counted all group elements or only distinct images. Multiple transformations can have the same effect on one point while differing elsewhere. The practice asks you to make that distinction explicit in both finite calculations and general proofs.

Continue

Conjugation is an action of a group on itself. Its stabilizers are centralizers, and its orbit partition becomes the class equation, a powerful source of divisibility arguments.