Cosets of $\mathbb Z \times (\mathbb Z \setminus 3 \mathbb Z)$ I need to find the cosets of
$$G =\mathbb Z \times (\mathbb Z \setminus 3 \mathbb Z)$$
With respect to this subgroup:
$$H = \{0 \} \times (\mathbb Z \setminus 3 \mathbb Z)$$
Where the operation on the latter group is addition modulo $3$. 
According to the definition of a coset, I need to take all elements of my initial group and multiply each of them by every element of $H$, so that I get a new set. The set is the coset I am looking for. 
Therefore, I take an arbitrary element $g \in G$:
$$g = (z, c) \text{ where }z \in \mathbb{Z} \text { and } c \in \{0,1,2\} $$
$$gH = (z,c) \cdot (0, d) \text{ where } d\in \{0,1,2 \} =(z+0,  (c+d)  \% 3) = (z, c)$$
And so, $gH$ = $\mathbb Z \times \{0,1,2\}$ 
Where $\%$ is the modulo operator. 
Is it the correct approach to solving this? 
 A: Yes, your approach is the right one.

A few comments regarding notation:


*

*When working with abelian groups, it is customary to denote the group operation with $"+"$ rather than $"\cdot"$.

*It is good notation to distinguish the elements of $\mathbb{Z}/3\mathbb{Z}$ using bars over them, that is, by writing $$\mathbb{Z}/3\mathbb{Z} = \{ \bar{0}, \bar{1}, \bar{2} \}$$ rather than $$\mathbb{Z}/3\mathbb{Z} = \{ 0,1,2 \}.$$ This is because in the latter case, it might be possible to confuse the elements with the regular integers.



One mistake in your final step (along with another comment on notation):


*

*The coset $gH$ is (also) a set, so it should be represented as such. I would write the computation you have given in the following manner:
$$
\begin{align}
gH &= (z,\bar{c}) + \{ (0,\bar{d}) : d \in \{0,1,2\} \} \\
&= \{ (z + 0,\bar{c} + \bar{d}) : d \in \{ 0,1,2 \} \} \\
&= \{ (z,\overline{c + d}) : d \in \{ 0,1,2 \} \} \\
&= \{ (z,\bar{d}) : d \in \{ 0,1,2 \} \}.
\end{align}
$$
So, $$gH = \{z\} \times \{ \bar{0}, \bar{1}, \bar{2} \} = \{z\} \times \mathbb{Z}/3\mathbb{Z}.$$ Note that this answer is different from what you have written! I hope it is clear where there was a mistake in your reasoning.


If anything is unclear in what I've written then feel free to ask in the comments below.
