(Group theory, group acting on set)

**Theorem :** if a group **G** acts on a set Ω then for each element **ω** in **Ω** -

**|Orb(ω)| . |Stab(ω)| = |G|**

What this theorem is saying (in a loose sense) is that for each element in **Ω**, **G** can be "factored through" by the actions which fix the element, and what's left corresponds to the elements which **G** takes our element to.

Note that this applies to infinite groups as well as finite ones, via the laws of arithmetic for cardinal numbers.

**Proof :** An application of Lagrange's theorem. Although clearly there are **|G|** possibilities for **ωg**, they are not necessarily distinct. But

**ωg**_{1} = ωg_{2}

<==> ωg_{1}g_{2}^{-1} = ω

<==> g_{1}g_{2}^{-1} in Stab(ω)

<==> Stab(ω)g_{1} = Stab(ω)g_{2}

So the different **ωg**'s correspond to the different cosets of **Stab(ω)**, and by Lagrange's theorem there are **|G| / |Stab(ω)** of them.

**Alternative proof** using coset spaces - pick an **ω in Ω**. Then **Orb(ω)** is a transitive G-space, and hence is isomorphic to **(G : Stab(ω))** (as explained and proved in coset space). And by a direct application of Lagrange's theorem, **|(G : Stab(ω))| = |G|/|Stab(ω)|**.