Select Page

Question:

How many subgroups does the group $$\mathbb{Z\times Z}$$ have?

A. 1

B. 2

C. 3

D. infinitely many.

Discussion:

How many subgroups does $$\mathbb{Z}$$ have? Well, for every $$m\ge 0$$ we have a subgroup $$m\mathbb{Z}$$.

And these are all distinct. So there are infinitely many subgroups of $$\mathbb{Z}$$.

So how many subgroups does $$\mathbb{Z}\times \{\bar{0}\}$$ have? Again, for every $$m\ge 0$$ we have a subgroup $$m\mathbb{Z}\times \{\bar{0}\}$$. That is again infinitely many.

Since $$\mathbb{Z}\times \{\bar{0}\}$$ is a subgroup of $$\mathbb{Z\times Z}$$ we have infinitely many subgroups of $$\mathbb{Z\times Z}$$.