I would just like to ask for a quick explanation on how one actually proves a group is cyclic and how to find the generator functions.
So the questions says:
Let $G_1$ be a group defined on the set $\mathbb Z_4$ with binary relation $+$ (addition modulo $4$), and let $G_2$ be a group defined on the set $\mathbb Z_5 − {[0]}$ with binary relation $\cdot$ (multiplication modulo $5$).
Prove that both $G_1$ and $G_2$ are cyclic and write down generators for each group.
I started by writing out the respective modulo tables for each group.
So for $(\mathbb Z_4 , +)$ you have the elements $\{0,1,2,3\}$ and for $(\mathbb Z_5 - {[0]}, \cdot)$ you have the elements $\{1,2,3,4\}$.
As far as I am aware, a cyclic subgroup of a group $(G , \star)$ is a set $\langle a \rangle = \{a^k : k \in \mathbb Z\}$, which is equal to $\{\dots, a^{-2}, a^{-1}, a^0, a^1, a^2, \dots \}$ and we call this the set generated by $a$.
I have the tables and definitions, but where do I go from here?
As a general rule, you prove a group is cyclic by exhibiting a generator. So you find an element $a\in G$ such that $G=\{a^n; n\in \Bbb{Z}\}$ or $G=\{na;n\in\Bbb{Z}\}$ when the group is noted additively.
So for $(\Bbb{Z}_4,+)$, $1$ is definitely a generator because
$$\{0,1,2,3\}=\{1,1+1,1+1+1,1+1+1+1\}$$
and for $(\Bbb{Z}_5,\cdot)$, $2$ is definitely a generator because
$$\{1,2,3,4\}=\{2,2^2,2^3,2^4\}$$