TITLE: Interesting Property of $(\Bbb Z_n,+)$
QUESTION [10 upvotes]: I've just started learning about groups in abstract algebra and today we were discussing subgroups. One of the examples we used in class was $(\Bbb Z_6,+)$, or integers mod six under addition. In addition to the trivial and improper subgroups, we found two additional subgroups:
$$\{0,1,2,3,4,5\}$$
$$\{0,2,4\}$$
$$\{0,3\}$$
$$\{0\}$$
What I found particularly interesting is that when looking at the cardinality of all four subgroups, they directly corresponded to the four factors of six. I tried thinking of other examples and each time it was the same story. What I want to know is if this generally holds for all $\Bbb Z_n\;\;(n\in\Bbb N)$, and if so, why it holds and what this means.

REPLY [4 votes]: This is a great example of Lagrange's Theorem! Simply stated, the theorem is 

For a finite group $G$ and a subgroup $H<G$, the order of $H$ divides the order of $G$.

Now, why should this be? The main idea is that you can construct an equivalence relation $\sim$ on $G$ where $a\sim b$ if and only if $a=bh$ for some $h\in H$.
Try to prove Lagrange's theorem using this relation!