I'm having trouble understanding the motivation behind a coset. The book I'm using (A Book of Abstract Algebra) states:
Let G be a group, and H a subgroup of G. For any element a in G, the symbol
denotes the set of all products ah, as a remains fixed and h ranges over H. aH is called the left coset of H in G.
It goes on to say the same about right cosets. I understand this definition (or I think I do), but what does this accomplish? Is it saying that given a subgroup H, you can generate a group G using cosets?
Thank you in advance.