Definitions: The order $|G|$ of a group $G$ is the number of elements of $G$. The exponent of a group is an integer $n$ such that $x^n = e$ for all $x\in G$ ($e$ is the neutral element).
Particularly, the definition doesn't say it's the least integer with the property. So in this definition the 4-group $V$ has exponent $4$ as well as exponent $2$.
But I'm having a hard time proving the following statement:
Lagrange's Theorem shows that a finite group $G$ of order $n$ has exponent $n$.
I know it must be very simple because it's not even stated as a corollary of Lagrange's.