Can anyone explain to me why $(34)(123) = (124)$? $$\begin{align}
(34)H&=  \{(34)(1),(34)(123),(34)(132)\}\\
& = \{(34),(124),(1432)\}.
\end{align} $$
Can anyone explain to me why $(34)(123) = (124)$?
I don't understand coset multiplying can you help me with this

If $\sigma=(123)$, $\tau=(34)$, and we compose right-to-left (as is consistent with your other compositions), then
$$\begin{align}
1 &\stackrel{\sigma}{\mapsto} 2 \stackrel{\tau}{\mapsto}2, \\
2&\stackrel{\sigma}{\mapsto} 3 \stackrel{\tau}{\mapsto} 4, \\
4 &\stackrel{\sigma}{\mapsto} 4 \stackrel{\tau}{\mapsto} 3, \\
3 &\stackrel{\sigma}{\mapsto} 1 \stackrel{\tau}{\mapsto} 1,
\end{align}$$
so $\tau\sigma=(34)(123)=(1243)$.
