the question about $6\mathbb{Z}^{2}$

Question

What do they mean when they write $6\mathbb{Z}^{2}$?

$6\mathbb{Z}^{2}=3\mathbb{Z}\times 2\mathbb{Z}$

$6\mathbb{Z}^{2}=6\mathbb{Z}\times \mathbb{Z}$

$6\mathbb{Z}^{2}=6\mathbb{Z}\times 6\mathbb{Z}$

Answer 1

They mean the third one, $6 \mathbb Z \times 6 \mathbb Z$. For good measure, let's understand this notation a bit more.

For an abelian group $A$, $6A$ denotes $\{6a : a \in A\}$, a subgroup of $A$. A priori, we therefore have two interpretations for $6\mathbb Z^2$

1) $(6\mathbb Z)^2$

2) $6(\mathbb Z)^2$

But per the above definition, these are the same subgroup of $\mathbb Z^2$.