[Math] Determining if the binary operation gives a group structure

Question

Let * be defined on $2\Bbb{Z}=\{2n\mid n\in \Bbb{Z}\}$ by letting $a * b=a+b$.

I've managed to determine that the operation is closed under $*$ and is associative. It's determining if the operation has an identity element and an inverse element that's the problem.

Here's my solution for the identity element:

Assume that the operation has an identity element $e\in G$ and let $a, b\in 2\Bbb{Z}$ such that $a*b=a$. Consider the element $-b$. It follows that $a+b+(-b)=a$.

I'm not sure how I'd find an inverse element and I'm not sure if my solution of the identity element is right.. Please advise.

Answer 1

Well, consider $2\mathbb Z$ as a subgroup of $\mathbb Z$. Then, what can you say?