So, I'm reading in Algebra by Serge Lang and I have come to the definition that the index of H in G, (G:H) is the number of left cosets of H in G. I'm having trouble figuring this out.
Take for example,
G = Z (the set of integers)
H = 2Z
Now, I know that (G:H) = 2 from various examples, but I don't understand why. What are the two cosets of 2Z that form H?
By my definition, a coset of H is a subset of G of form aH where a is from G. But for any a in Z, a(2Z) = 2Z. So how do we get two cosets?