This a followup question on the first answer to this post:
Why the term and the concept of quotient group?
In the first answer, Lahtonen says that for the quotient group Z/10Z,
one can "equate 9 with 99", etc.
Let's take Z/3Z to make the question shorter. The quotient group has three elements:
(…,0,3,6…),(,1,4,7…),(,2,5,8…)
Each element is an infinite set of integers, specifically, the integers divisable by 3 plus
an offset that is 0,1, or 2.
From my understanding of the answer by Lahtonen (which I'm not questioning, I've seen the same idea stated somewhere else) it should be that these cosets are somehow the same as equating 3 and 0, or 5 and 2.
The point is lost on me. In what sense does "5==2" relate to the 3 cosets?
The latter are infinite, and have more structure than the single integers.
Best Answer
Let $G$ be a group and $N$ be normal in $G$. So $G/N$ is in fact studying elements outside of $G$. Just think $N$ is now our identity element. Elements in $N$ are no longer interesting. Furthermore two elements are equivalent if they differ by an element in $N$. Taking the quotient $\mathbb{Z}/3\mathbb{Z}$ is not saying $2==5$ all the time. It is more saying $2$ and $5$ differ by something in $3\mathbb{Z}$.