 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/2015/prelims/mathematics-i/7/a/ii/solution

s

with the set of left cosets of

G_{s}

. Explicitly, there is a bijection

G / G_{s} ⟶ G \cdot s, g G_{s} ⟼ g \cdot s .

Here

G / G_{s}

denotes a set of cosets; it is not asserted to be a quotient group. To prove the identification, observe that

g G_{s} = h G_{s} ⟺ h^{- 1} g \in G_{s} ⟺ (h^{- 1} g) \cdot s = s ⟺ g \cdot s = h \cdot s .

This equivalence proves both well-definedness and injectivity. Every element of the orbit of a group action has the form

g \cdot s

In particular, for a finite group, the cardinalities satisfy

∣ G \cdot s ∣ = [G : G_{s}] = \frac{∣ G ∣}{∣ G _{s} ∣}, ∣ G ∣ = ∣ G \cdot s ∣ ∣ G_{s} ∣.

The last equality follows from Lagrange's theorem applied to the stabilizer subgroup.

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