Yes, it is.
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Yes.
No. An addition operation need not even be defined.
Yes, they are.
Yes. The entire set of natural numbers is closed under addition (but not subtraction). So are the even numbers (but not the odd numbers), the multiples of 3, of 4, etc.
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.