 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2022/paper-2/9/b/solution

For each

g

, the map

L_{g} : H ⟶ g H, h ⟼ g h

is surjective by definition of a coset and injective by left cancellation. Its inverse is multiplication by

g^{- 1}

. Therefore

∣ g H ∣ = ∣ H ∣ for every g \in G .

The same argument with right multiplication proves the analogous result for right cosets. This equality of sizes is the counting step in Lagrange's theorem.

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