Permutation group isomorphism

Question

I've already proved H is a subgroup as it was very straight forward and easy. I'm certain there is an extremely obvious and easy isomorphism that I am somehow missing. Thanks for any help.

Answer 1

Define $\phi:S_{n-1}\to S_n$ as follows: If $\sigma\in S_{n-1}$, let $\phi(\sigma)\in S_n$ be the map $\phi(\sigma)(k)=\sigma(k)$ for $k<n$ and $\phi(\sigma)(n)=n$.

Answer 2

The subgroup $H$ has index $n$ is $S_n$, so must be isomorphic to $S_{n-1}$, as we have seen here. But of course, it is much more straightforward to write down an isomorphism.

Answer 3

For any $\sigma\in H$, consider its restriction to the set $\bigl\{1,\dots, n-1\bigr\}$.

This restriction of $\sigma$ takes its values in $\bigl\{1,\dots, n-1\bigr\}$, by definition of $H$, so restricting its codomain to $\bigl\{1,\dots, n-1\bigr\}$ makes it a permutation of $S_{n-1}$.