I'm working on some released exams, in particular this one and I am stuck on 4B.
I know I need to prove three properties to show that it is a group:
- The operation is associativity
- There exists an identity
- There exists an inverse.
I can find a solution for 2 and 3 somewhat easily:
For 2:
Let $x$ be an element belonging to the equivalence class. Then $[x + 0] = [x]$, which implies $[0]$ (not actually zero itself, but zero's equivalence class) is the identity element.
For 3:
Let $[x]$ be an element belonging to the equivalence class. Then $[x + (-x)] = [0]$ which implies $[-x]$ is the inverse of [x].
I am having trouble with the last part, associativity. I'm virtually lost:
Let $x,y$ and $z$ be elements of arbitrary equivalence classes. Then:
$$[[x + y] + z] = ?$$
I don't know what to do from here. Any ideas? Also, when the question states:
Be sure to check that the operation is well defined.
I don't understand what I am supposed to be looking for. My first though is that there isn't a unique identity or inverse... But the more I stare at the problem, I believe there is.