3. Let m and n be positive integers such that m divides n. (a) Prove that the map y: Z/nZ Z/mZ sending a + nZ to a + mZ for any a E Z is well-defined. (b) Prove that y is a surjective group homomorphism. (c) Determine the group structure of the kernel of p.

Given m and n are two positive integers such that m divides n.

To prove the map $\phi :\mathrm{\mathbb{Z}}/n\mathrm{\mathbb{Z}}\to \mathrm{\mathbb{Z}}/m\mathrm{\mathbb{Z}}$ such that $\phi \left(a+n\mathrm{\mathbb{Z}}\right)=a+m\mathrm{\mathbb{Z}}$ for any $a\in \mathrm{\mathbb{Z}}$ is well-defined

Let us consider two integers $a$ and $a\text{'}$ such that

 $a+n\mathrm{\mathbb{Z}}=a\text{'}+n\mathrm{\mathbb{Z}}$                       (i)

Ten, eq. (i) implies $n$ divides $\left(a-a\text{'}\right)$. Now, according to the definition of $\phi$, 

$\phi \left(a+n\mathrm{\mathbb{Z}}\right)=a+m\mathrm{\mathbb{Z}}$

$\phi \left(a\text{'}+n\mathrm{\mathbb{Z}}\right)=a\text{'}+m\mathrm{\mathbb{Z}}$

Since $n$ divides $\left(a-a\text{'}\right)$ and $m$ divides $n$, by transitivity, $m$ divides $\left(a-a\text{'}\right)$. So $a+m\mathrm{\mathbb{Z}}=a\text{'}+m\mathrm{\mathbb{Z}}$ implying 

$\phi \left(a+n\mathrm{\mathbb{Z}}\right)=\phi \left(a\text{'}+n\mathrm{\mathbb{Z}}\right)$

Hence $\phi$ is well-defined