$H=${${h^k}$|$k\in Z$} for some $h$ in $H$.
(a) If $G$ is any group, and $g$ is a particular element of $G$, show that the set {${g^k}|k \in Z$} is a subgroup of $G$: The set of all integral powers of given element of a group is a cyclic subgroup.
(b) Let {$1,2,3,4$} be the set of multiplicative group of the field $Z/(5)$. Show that this group is cyclic by showing that 2 is a generator of the group. Show that 3 also generates the group.
I was taught about groups for 10 minutes in my class and then ran out of time, yet I've been ask to study for this for an exam. However, reading up on group and cyclic groups, I'm never getting far with this question, I know cyclic groups, there is one generator that can generate the entire group of elements within that group. I just don't know how to show that it's a subgroup for (a). For (b), how do I show $2$ and $3$ are generator of that field?
For (a) you can perform the one-step subgroup text: If $a,b \in H$ implies $ab^{-1}$ in $H$ then $H$ is a subgroup.
So let $g^k$ and $g^j$ be in $H=\{g^i \mid i \in \mathbb Z\}$. Then $g^kg^{-j} =g^{k-j}$ is in $H$. Hence $H$ is a subgroup of $G$.
For (b) write out the powers of $2$:
$2^0 = 1, 2^1 = 2, 2^2 = 4, 2^3 = 3, 2^4 =1$. Hence $2$ generates $\mathbb Z / 5 \mathbb Z$.
Do the same with $3$ to show that $3$ generates $\mathbb Z / 5 \mathbb Z$.