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 Incoming links: Past exam of the mathematics course of the University of Oxford / 2021 / Prelims / Mathematics I / 6 / a / i

Past exam of the mathematics course of the University of Oxford / 2021 / Prelims / Mathematics I / 6 / a / iii / Solution 2026-10-25

Let the finite set

G / H

contain

r

distinct left cosets. By i, they form a disjoint partition of a set

G

, and by ii each has

∣ H ∣

elements. If

H

is also finite, counting this finite disjoint set union gives

∣ G ∣ = W \in G / H \sum ∣ W ∣ = r ∣ H ∣ = ∣ G / H ∣ ∣ H ∣.

Thus

G

is a finite group. This is the counting argument behind Lagrange's theorem.

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