Let $n,k \in \mathbb{N}$ with $1<k<n$. We define a subgroup $H \leq S_n$ as:
$$H=\{\sigma \in S_n | \; \sigma(\{1,…,k\}) \subseteq \{1,…,k\}\}$$
I have to show that $H$ is isomorphic to $S_k\times S_{n-k}$.
My idea is that for all permutations in $H$, we can divide it in two parts, one of them only permutates elements from $\{1,…,k\}$ and the other one permutates only elements from $\{k+1,…,n\}$, so I could write $\sigma \in H$ as $\sigma = \rho\tau, \rho \in S_k$ and $\tau \in S_{n-k}$.
With that, an isomorphism should be easy to define, but I don't really know how can I formalize that idea (or if it's really right). Any hints? Thanks.
Best Answer
Deploying my hint in the comment, let's define $I:=\{1,\dots,k\}$ and $J:=\{k+1,\dots,n\}$:
and $H \cong S_k\times S_{n-k}$.