Example for Cyclic Groups and Selecting a generator In Cryptography, I find it commonly mentioned: 
Let G be cyclic group of Prime order q and with a generator g.
Can you please exemplify this with a trivial example please!
Thanks.

In fact if you take the group $(\mathbb{Z}_{p},+)$ for a prime number $p$, then every element is a generator. 


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*Take $G= \{a^{q}=e, a ,a^{2}, \cdots, a^{q-1}\}$. Now $|G|=q$ and $G = <a>$, which means that $G$ is generated by $a$.

