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?
Really, that's bad notation but a necessary evil.
You'll remember that a group is a non-empty set along with an associative binary operation (function) such that there is some with for all . For each there is some such that . And closure (but that is implicit in how we defined .
That said NO ONE wants to keep writing everytime. So, instead of we write . This does not mean multpilcation. This is just short hand notation for the function . Remember, can be many, many things (function composition, multplication, matrix multiplication, addition, etc.). Now, you'll also remember that in general and groups for which all the elements satisfy that aer called abelian. Now, it is customary (don't ask me why) to switch from using the short hand notation to even though it means the same thing when the group is abelian.
So, it is true that a coset is of the form but that really means and for the group we have that means addition.