I am wondering if anyone can elaborate on the following Wikipedia example of a coset.
I am confused how the quotient group is even a group in this example. I will try to explain each step of what the example does and hopefully my mistakes will help anyone to help me learn where my problems are...
We are considering $G=(\mathbb{Z}_6,+)=\{0,1,2,3,4,5\}$ and the subgroup $N=\{0,3\}$. We want to form the set of left cosets (I know N is normal, so its left and right cosets are equivalent).
The Wikipedia article begins to construct the set $G/N=\{gN\mid g\in G\} = \{g\{0,3\}\mid g\in \{0,1,2,3,4,5\}\}$
One particular coset with $g=0$ is $\{\{0+0\equiv 0\pmod{6}\},\{0+3\equiv 3\pmod{6}\}=\{0,3\}$. Once we construct all six cosets, we eliminate the three equivalent sets and we're left with a set of 3 unique cosets: $G/N=\{\{0,3\},\{1,4\},\{2,5\}\}$. To me this doesn't seem like a group. The elements of this "group" are sets like $\{0,3\}\in G/N$. Another element is $\{1,4\}$. How do I show that $\{0,3\}+\{1,4\}$ is closed and is an element of $G/N$?
Thanks for your help. I hope you don't mind clearing up my misunderstandings.