A group isomorphism between $\mathbb{Q/Z}$ and $\mathbb{Q/2Z}$

Question

Question: Prove that $\mathbb{(Q/Z, +)}\cong\mathbb{(Q/2Z, +)}$

My attempt To prove they are isomorphic I need to define a map from $\mathbb{Q/Z}$ to $\mathbb{Q/2Z}$ which is bijective and preserve the group operation.

On a first sight I thought, $q+\mathbb{Z}\mapsto q+\mathbb{2Z}$ will work. But, it doesn't work (i think). Then I think that, can we use Fundamental theorem of group homomorphism? That is, If we define map $f:\mathbb{Q}→\mathbb{Q/2Z}$ which is homomorphism with kernel $2\mathbb{Z}$ then, it will do the job. But I am stuck here, unable to define such a map.... please help.

Answer 1

The correct bijection is $$q+\mathbb Z\mapsto2q+2\mathbb Z$$ and your proposal does not work because it has no preimage for (say) $1+2\mathbb Z$.