What does $\Bbb{Z}_{18}$ mean in this question? I am trying to solve this question

Let $G=\Bbb{Z}_{18}$ and $H$ a subgroup of G generated by $3$, Talk about normal subgroups and coset multiplication then find the cosets $(G/H,\bigotimes)$

Does $\Bbb{Z}_{18}$ mean $$\{0,1,2,3,...,16,17\}$$
or
$$\{1,a,a^2,...,a^{16},a^{17}\}$$
where $a$ is the generator of $G$.
If it is the second the definition then what does $a$ equal to here?
I am just confused if I am supposed to use the multiplication or addition here.
 A: As I am looking Your first option sounds good to me. if your $Z_{18}$ is $\Bbb{Z}_{18}$.
and the cosets in  $\Bbb{Z}_{18}/H$ are
$0+H=\{0,3,6,9,12,15\}$,
$1+H=\{1,4,7,10,13,16\}$  and so on. Can you complete it?
A: Both are same. In the second group if you replace a by $1$ and use the operation of addition modulo $18$, then you get $\mathbb{Z}_{18}$. The notation $a^n$ means you apply the group operation on $a$, $n$ times. Both the sets you mentioned will be isomorphic if you use the map $1$ goes to $a$. So both are isomorphic. We generally use the first set you mentioned under addition modulo $18$ as it is easier to handle.
If you are talking about the normal subgroups of $G/H$:
As far as the quotient is concerned, $G/H$ has three elements and hence is a cyclic group (thus abelian) of order $3$. It is of the form $\{ H, 1+H, 2+H \}$. Since it is abelian, all the subgroups are normal. But since the order of group is $3$ it has only two subgroups $\{e\}$ (where $e$ is the identity) and the group $G/H$ itself.
A: $\mathbb{Z}_{18}$ is the set of integers mod $18$. It is a group of order $18$ defined under modulo addition.
$$\mathbb{Z}_{18} = \{0, 1, 2, 3,..., 17\}$$
Notice that we stop at $17$. This is because $0$ is the identity element. If you're not familiar with modulo arithmetic, you may also be confused about how this group is closed under addition. After all, isn't $1+17=18$? Well, in this group, $17$ is the inverse of $1$. So $1+17=0$. Furthermore, $$2+16=0$$ $$3+15=0$$
and so on.
To answer your second question, factor (quotient) groups are defined under coset multiplication, which is defined as follows:
$$(aH)(bH) = (a\cdot b)H$$
where $\cdot$ is the binary operation of the original group.
