Left cosets are all in bijection

Written by Patrick Stevens last updated

Let be a subgroup of . Then for any two left cosets of in , there is a bijective function between the two cosets.

Proof

Let be two cosets. Define the function by .

This has the correct codomain: if (so , say), then so .

The function is injective: if then (pre-multiplying both sides by ) we obtain .

The function is surjective: given , we want to find such that . Let to obtain , as required.

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