coset

For a group \(G\) with subgroup \(H\), the left coset corresponding with element \(g\) of \(G\) is \(\{gh | h\in H\}\).

THe right coset is \(\{ hg | h\in H\}\)

For a normal subgroup, the left and right coset for a \(g\in G\) are the same.

Then, to find all cosets, we can let \(g\) range over all of \(G\) and collect the resulting cosets. It turns out among the generated cosets, any two cosets will either be identical or disjoint

Also, the cosets will cover all elements in \(g\in G\), because \(e \in H\), so \(g\in gH\).