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I am doing self study of Abstract Algebra from Pinter.

My doubt is regarding Chap 13 Counting Cosets:

A coset contains all products of the form "ah" where a belongs to G and h belongs to H where H is a subgroup of G. So each coset should contain the number of elements in H. Now the number of cosets should be the number of elements in G since each element of G is used to construct a coset. So the number of elements in all cosets should be number of elements in G*Number of elements in H - But the family of cosets is a partition of G and should have the same number of elements of G... there is definitely something wrong in the second-third line of this argument... but I am not able to pin it down