(TIFR 2014 problem 16)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}).
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}).
