We looked briefly at this example in class but I'm not quite sure how to proceed, and I can't find examples of this in any textbooks I have (Dummit & Foote and Nicholson).
Suppose we have $H = \langle(1,1) , (1,-1)\rangle \le G = \mathbb{Z}^2$ for groups $H$ and $G$. Find $|G:H|$.
I think I'd have to take the standard basis for $G$ and then express the elements in $H$ as some combination of this basis. The goal (from what I understood, at least) seems to be to express $G$ and $H$ as direct products and then look at the order of the quotient (since that's equal to $|G:H|$).
Am I on the right track here? How would I go about actually showing all the work for this question? Thanks for reading.