This was a bonus question on my old homework assignment and so I didn’t receive a solution for this.
Let $G\leq S_m$ and $H\leq S_n$ be permutation groups. Show that there exists a permutation group $K\leq S_{m+n}$ for which $G\times H\cong K$.
I don’t really have a clue on what kind of observations are needed to even get started. I have just tried to see why this would be true for some m,n but I dont have an intuition for what K would look like. Thanks for the help!
Hint: There is a inclusion $S_n \times S_m \hookrightarrow{} S_{n+m}$ by defining $\sigma \in S_n$ to act on the first $n$ elements of $\{1, \dots, m+n\}$ and $\tau \in S_m$ to act on the last $m$ elements.