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

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- Solution 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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