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?

  • $\begingroup$ Don't you mean external direct sum? If so, then the external direct sum is isomorphic to the internal direct sum. $\endgroup$ – Alvin Lepik Nov 26 '17 at 17:02
  • $\begingroup$ I am using Gallian's text, and he refers to what I have written above as the external direct product. $\endgroup$ – 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 $(1,1) \in \mathbb{Z}_{n_1} \times \mathbb{Z}_{n_2}$


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.