[Math] Does set of rational numbers with odd denominators form a group with binary operation of $+$

Question

Does set of rational numbers with odd denominators form a group with
binary operation of $+$?

I think no, because it doesn't have an identity element since $0$ is not in the set because it doesn't have a denominator.

Can I say $0$ does not have a denominator, so it doesn't have an odd denominator?

Answer 1

Yes, it forms a group. It has an identity element, which is $0\left(=\frac01\right)$.