How would one compute the order of $G=\mathbb{Z}^*_{13}\times\mathbb{Z}^+_{13}$. (Where $\mathbb{Z}^+$ denotes the additive group.) And is $G$ then cyclic?
This is probably very straight forward, but for some reason I am stumped on how to compute this.
Hint:
(i) If $G=H\times K$, then $|G|=|H||K|$.
(ii) Let $G=H\times K$ where $H$ and $K$ are cyclic groups of order $m,n$ respectively, then $G$ is cyclic if and only if $m$ and $n$ are relatively prime.
(iii) $|\Bbb{Z}_n^*|=\phi(n)$ where $\phi(n)$ is the Euler phi function, which is the number of positive number which are relatively prime to $n$. In particular if $n$ is prime, then $\phi(n)=n-1$. Also $\Bbb{Z}_n^*$ is cyclic if and only if $n$ is prime.