Is $\mathbb{Q} /\mathbb{2Z}$ is not isomorphic to $\mathbb{Q}/ \mathbb{5Z}$ ?
How can i check this? the elements of $\mathbb{Q} /\mathbb{2Z}$ is be of form
$$\frac{p}{q} +2\mathbb{Z}$$ where $p$ and $q$ are Integers. similarly the elements in $\mathbb{Q}/ \mathbb{5Z}$ is of form $$\frac{p'}{q'} +5\mathbb{Z}$$
I have no idea how can i proceed it further, can i draw a isomorphism form $\mathbb{Q} /\mathbb{2Z}$ to $\mathbb{Q} /\mathbb{5Z}$?
Please Help!
Thankyou.
Hint: You have an isomorphism $$\varphi:\mathbb{Q} \rightarrow \mathbb{Q}, x\mapsto \frac{5}{2}x.$$
Now check that this extends to an isomorphism between your desired groups. Indeed, you can compose $\varphi$ with the projection onto $\mathbb{Q}/5\mathbb{Z}$, compute the kernel and use the correct isomorphism theorem for groups to conclude.