# Finding cosets of a quotient group: List the cosets of $HN/N$

In the group $\Bbb Z_{24}$, let $H=\langle 4\rangle$ and $N=\langle 6\rangle$

1) List the elements of $HN$. I found $HN=\{0,2,4,\cdots,22\}=\langle 2\rangle$

2) List the elements of $H\cap N$. I found $H\cap N=\{0,12\}=\langle 12\rangle$

3) List the cosets of $HN/N$. How do I do this?

Simply list the cosets. Two cosets $a+N$, $b+N$ will be the same if $a-b$ is divisible by $6$. $$0+N=N$$ $$2+N=\{2,8,14,20\}$$ $$4+N=\{4,10,16,22\}$$ We know there are only $3$ since $|HN|/|N|=3$.