TITLE: Exponent of a finite abelian group
QUESTION [5 upvotes]: I have a very basic question:
Let $G$ be a finite abelian group and let $m$ be the exponent of $G$. 
Then does there exist $g\in G$ s.t. o$(g)=m$ and if so, why?
Many thanks in advance.

REPLY [10 votes]: Yes.
The reason is that if $n$ is the largest among the orders of elements of $G$, and $g$ is an arbitrary element of $G$, then the order of $g$ divides $n$.
To see this, one can show that if $g$ and $h$ have orders $m$ and $k$ then there is a product of the form $g^{m'}h^{k'}$ with order $\rm{lcm}(m,k)$ (pick $m'$ and $k'$ such that the orders of $g^{m'}$ and $h^{k'}$ are coprime and as large as possible).