TITLE: Largest order of an element in a group
QUESTION [7 upvotes]: How to determine the largest order of an element in the group $\mathbb Z_4 \times \mathbb Z_{18} \times \mathbb Z_{15}$. 

I know the order of the element $(a, b)$ in the direct product $\mathbb Z_m \times \mathbb Z_n$ is $\operatorname{lcm}(|a|, |b|)$. How to apply this result here? Thanks for the help.

REPLY [7 votes]: Using the Fundamental Theorem of Finite Abelian Groups you may write $$\mathbb{Z}_4\times \mathbb{Z}_{18}\times \mathbb{Z}_{15}\cong\mathbb{Z}_4\times (\mathbb{Z}_9\times\mathbb{Z}_2)\times(\mathbb{Z}_3\times\mathbb{Z}_5)\cong (\mathbb{Z}_3\times \mathbb{Z}_9)\times (\mathbb{Z}_2\times\mathbb{Z}_4)\times (\mathbb{Z}_5)\cong \mathbb{Z}_6\times \mathbb{Z}_{180}$$
The last is written in canonical form, where each order divides the next higher one.  Once this is done the largest order of any element (called the exponent of the group) is always the largest order, in this case $180$.