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Say we have a group G, and some subgroup H. The number of cosets of H in G is called the index of H in G. This is written [G:H].
If G and H are finite, [G:H] is just |G|/|H|.
What if they are infinite? Here is an example. Let G be the integers under addition. Let H be the even integers under addition, a subgroup. The cosets of H in G are H and H+1. H+1 is the set of all even integers + 1, so the set of all odd integers. Here we have partitioned the integers into two cosets, even and odd integers. So [G:H] is 2.
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