Cosets of $\mathbb Z \times (\mathbb Z \setminus 3 \mathbb Z)$

Question

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?

Answer 1

Yes, your approach is the right one.

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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.

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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.