Let $H = \{(1),(12)(34)\}$ and $\alpha_1 = (243), \alpha_2 = (142), \alpha_3 = (132),$ and $\alpha_4 = (234)$.
Coset multiplication is a bit confusing to me. The book states in an example that $\alpha_1H = \alpha_2H$ and I'd like to see it for myself, however not a single paragraph in this book actually explains the operation of $\alpha_1H$. As to why I wrote the other $\alpha_i$'s, the book claims that:
$\alpha_1 \alpha_3H \neq \alpha_2 \alpha_4H$
I just would like to see it for myself before I press forward.
Thanks!