Okay, I'm trying to understand exponents of groups.
I will start with the Set $Z_3$, where $Z_3$ is the integers mod3 under addition.
Now, I want to set out to find the exponent of this group, but lets first find the order of the elements.
The order of $1$ is $3$. This is because $1 = 1$, $1 + 1 = 2$, and $1 + 1 + 1 = 0$. The least positive integer $n$ for which $1^n = e$(our identity)$ = 3$.
This also holds for $2$.
But what about for our identity $0$? The least number $n$ such that $e^n = e$ is equal to $1$. What will be the exponent of this group then? Allegedly, all finite groups have an exponent, but this can't possibly have one because these numbers don't share anything in common. Am I missing something?