Orbit-stabiliser theorem

Let \(G\) be a finite group, act­ing on a set \(X\). Let \(x \in X\). Writ­ing \(\mathrm{Stab}_G(x)\) for the sta­bil­iser of \(x\), and \(\mathrm{Orb}_G(x)\) for the or­bit of \(x\), we have

$$|G| = |\mathrm{Stab}_G(x)| \times |\mathrm{Orb}_G(x)|$$
where \(| \cdot |\) refers to the size of a set.

This state­ment gen­er­al­ises to in­finite groups, where the same proof goes through to show that there is a bi­jec­tion be­tween the left cosets of the group \(\mathrm{Stab}_G(x)\) and the or­bit \(\mathrm{Orb}_G(x)\).

Proof

Re­call that the sta­bil­iser is a sub­group of the par­ent group.

Firstly, it is enough to show that there is a bi­jec­tion be­tween the left cosets of the sta­bil­iser, and the or­bit. In­deed, then

$$|\mathrm{Orb}_G(x)| |\mathrm{Stab}_G(x)| = |\{ \text{left cosets of} \ \mathrm{Stab}_G(x) \}| |\mathrm{Stab}_G(x)|$$
but the right-hand side is sim­ply \(|G|\) be­cause an el­e­ment of \(G\) is speci­fied ex­actly by spec­i­fy­ing an el­e­ment of the sta­bil­iser and a coset. (This fol­lows be­cause the cosets par­ti­tion the group.)

Find­ing the bijection

Define \(\theta: \mathrm{Orb}_G(x) \to \{ \text{left cosets of} \ \mathrm{Stab}_G(x) \}\), by

$$g(x) \mapsto g \mathrm{Stab}_G(x)$$

This map is well-defined: note that any el­e­ment of \(\mathrm{Orb}_G(x)\) is given by \(g(x)\) for some \(g \in G\), so we need to show that if \(g(x) = h(x)\), then \(g \mathrm{Stab}_G(x) = h \mathrm{Stab}_G(x)\). This fol­lows: \(h^{-1}g(x) = x\) so \(h^{-1}g \in \mathrm{Stab}_G(x)\).

The map is in­jec­tive: if \(g \mathrm{Stab}_G(x) = h \mathrm{Stab}_G(x)\) then we need \(g(x)=h(x)\). But this is true: \(h^{-1} g \in \mathrm{Stab}_G(x)\) and so \(h^{-1}g(x) = x\), from which \(g(x) = h(x)\).

The map is sur­jec­tive: let \(g \mathrm{Stab}_G(x)\) be a left coset. Then \(g(x) \in \mathrm{Orb}_G(x)\) by defi­ni­tion of the or­bit, so \(g(x)\) gets taken to \(g \mathrm{Stab}_G(x)\) as re­quired.

Hence \(\theta\) is a well-defined bi­jec­tion.

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  • Group action

    “Groups, as men, will be known by their ac­tions.”