Let $G=\mathbb{Z}\times\mathbb{Z}$ and $H=\{(a,b)\in\mathbb{Z}\times\mathbb{Z}: 8\mid a+b\}$. What is the index $[G:H]$?

Question

Let $G=\mathbb{Z}\times\mathbb{Z}$ and $H=\{(a,b)\in\mathbb{Z}\times\mathbb{Z}: 8\mid a+b\}$. What is the index $[G:H]$?

From a separate exercise part of this problem we are given $H\lhd G$ and that $G/H\cong\mathbb{Z}_8.$

Considering the definition of $H$ I believe $|H|=8n$ (Since there are $8n$ entries that satisfy the condition).

How would I find $|G|$? Would this just be $\infty$?

Answer 1

By definition, when we have $H\lhd G$ such that $G/H$ is finite, then $[G:H]:=|G/H|.$ In this case, then, $[G:H]=|\Bbb Z_8|=8.$

Answer 2

Well, if you already know that $G/H\cong\mathbb{Z}/8\mathbb{Z}$, then the index is $8$.

How can you prove the isomorphism? Just consider the obvious map $$ \varphi\colon\mathbb{Z}\times\mathbb{Z}\to\mathbb{Z}/8\mathbb{Z}, \qquad \varphi(a,b)=a+b+8\mathbb{Z} $$ and prove it is a homomorphism. It is clearly surjective and its kernel is $H$. Thus the homomorphism theorem provides the required isomorphism.