Properties of $\Bbb{Q}/n\Bbb{Z}$

Question

For $n$ a positive integer, let $\Bbb{Q}/n\Bbb{Z}$be the quotient of the group of rational numbers $\Bbb{Q} $ by the subgroup $n\Bbb{Z} $. For each of the following statements state whether it is true or false.
(a)every element of $\Bbb{Q}/n\Bbb{Z}$ is of finite order.
(b) There are only finitely many elements in $\Bbb{Q}/n\Bbb{Z}$ of any given finite order.
(c) Every proper subgroup of $\Bbb{Q}/n\Bbb{Z}$ is finite.
(d) $\Bbb{Q}/2\Bbb{Z}$and $\Bbb{Q}/5\Bbb{Z}$ are isomorphic as groups.
Try
(a) $\frac ab +n\Bbb{Z}$ be any element of $\Bbb{Q}/n\Bbb{Z}$.
Than $\frac ab +... +\frac ab (\text{adding nb times} ) =\frac {nab}b$ hence has finite order $nb$ (b) take $n>2$ and $n$ be a fixed prime. Then the element $\frac pn$ where $p$ is any prime, has order $n^2$ but there are infinitely many primes hence infinitely many such elements possible.
I am unable to proceed the other two options. Please help. Thanks in advance

Answer 1

$\mathbb Q/\mathbb Z\to\mathbb Q/n\mathbb Z:a\mapsto n\cdot a$ is an isomorphism, so we only need to consider the case when $n=1$ in these questions. (this also immediately answers (d).)

HINTS:

(b) Your approach is on the right track, you should say the elements of order $d$ can be expressed as $m/d$ for some integer $m$, so...

(c) This is not true. Let $A=\{m/2^k:m,k\in\mathbb Z\}\subseteq\mathbb Q$ and consider $A/\mathbb Z\subset\mathbb Q/\mathbb Z$ (i.e., the group of elements with denominator a power of $2$.)