I've recently been introduced to cosets as part of a group theory subsection of my Introduction to Pure Mathematics module, but no matter how many explanations I see online I can't seem to wrap my head around what the point of them is?
The example given in the lecture notes is as follows:
Suppose $G=S_{3}$ and $H$ is the following subgroup of order 2: $$H=\{ e,(1~2) \}$$ Then the right cosets of $H$ are: $$\begin{align} H(1~3)=\{ (1~3),(1~3~2) \}&&\text{and}&&H(2~3)=\{ (2~3),(1~2~3) \} \\ \end{align}$$ Whereas the left cosets are: $$\begin{align} (1~3)H=\{ (1~3),(1~2~3) \}&&\text{and}&&(2~3)H=\{ (2~3),(1~3~2) \} \end{align}$$
But why would I want to do that? I feel that I may be misunderstanding these because I don't fully understand the "backstory" to these, but that is much too long to include here and I'm not even sure that it's relevant, the lecture notes seem to be a bit of a mess and have a habit of describing things in a much more complicated manner than they actually are. I often find sets of notes from other universities that describe the same topic in a much simpler way, but I've yet to find something on cosets yet.
If I could be pointed in the direction of some resources to read or just someone else's explanation of what these are that would be greatly appreciated. Thank you in advance!
