The order of elements in the quotient group Let $G$ be a group, $N$ a normal subgroup of $G$, $a \in G$, and let $k = o(a)$.
I don't understand why the order of an element in $G/N$ is not necessarily equal to the order of the "corresponding" element in $G$ (i.e, why it might be that $o(a) \neq o(aN)$).
My reasoning is this: We know $k$ to be the smallest positive integer such that $a^k = e$. 
Let $m = o(aN)\Rightarrow (aN)^m=eN \Rightarrow (a^m)N=eN \Rightarrow a^m=e$.
If $m < k$, it is a contradiction to $k$ being $a$'s order.
If $m > k$, than $k$ is really $a$'s order since $(aN)^k=(a^k)N=eN=N$.
I know I'm wrong, but I am not sure where.

Recall that the cosets $aN$ are equivalence classes. When two cosets are equal, say $sN=tN$, it means both the representatives $s$ and $t$ are in the same equivalence class. It does not mean that the two are equal (they are simply equivalent). 
In particular, $sN=tN$ just implies $st^{-1} \in N$. This is not the same as saying $s=t$.
