# Proving an Element is of the Largest Order

I am trying to prove that $(1,1)$ is an element of largest order in $Z_{n_1} \oplus Z_{n_2}$, where $\oplus$ refers to the external direct product.

I am slightly confused because there is no restriction on $n_1$ or $n_2$ being distinct. For instance, wouldn't it be the case that $Z_{2} \oplus Z_{2}$ does not even have $(1,1)$ as an element?

• Don't you mean external direct sum? If so, then the external direct sum is isomorphic to the internal direct sum. – Alvin Lepik Nov 26 '17 at 17:02
• I am using Gallian's text, and he refers to what I have written above as the external direct product. – Nina Nov 26 '17 at 17:04

hint try to prove that the order of $(1,1)$ is always the lcm of $n_1,n_2$.
Notice that, $1 \in \mathbb{Z}_n$ for all $n \in \mathbb{N} \setminus \{ 0 \}$, So $(1,1) \in \mathbb{Z}_{n_1} \times \mathbb{Z}_{n_2}$