# A finitely generated abelian group $G$ has an element of order $\exp(G)$

Let $$G = \langle e_1, \dots, e_n \rangle$$ be a finitely generated abelian group. Denote by $$n = \exp(G)$$ the least common multiple of $$\{\operatorname{ord}(e_1), \dots, \operatorname{ord}(e_n)\}$$. Then $$n$$ is the LCM of the order of all elements of $$G$$.

How do you show that there is an element $$a \in G$$ of order n?

This should be done without using the fundamental theorem of finitely generated abelian groups, as I want to use this statement to prove a special case of the fundamental theorem.

For every prime power $$p^k$$ dividing $$n$$, we find $$e_j$$ such that $$p^k\mid \operatorname{ord}(e_j)$$, hence find a multiple with order $$=p^k$$. Verify that the sum of such elements for powers of different $$p$$ has order the product of the involved prime powers. If we take maximal prime powers, this product is the LCM.