Let $H=\{\beta \in S_5\mid\beta(1)=1\}$ and $K=\{\beta \in S_5\mid\beta(2)=2\}$. Prove that $H$ is isomorphic to $K$.

Question

Let $H=\{\beta \in S_5\mid\beta(1)=1\}$ and $K=\{\beta \in S_5\mid\beta(2)=2\}$. Prove that $H$ is isomorphic to $K$.

$H$ is the group of permutations fixing $1$ and $K$ is the group of permutations fixing $2$. I don't know how to write the isomorphism $\phi$.

Answer 1

Hint: $H$ is conjugated to $K$ by the transposition $(1,2)$.

Answer 2

It's not random that $H$ and $K$ are conjugated: they are the stabilizers of $1$ and $2$, respectively, under the transitive action of $S_5$ on $\{1,2,3,4,5\}$ (as a group of permutations).

Another approach: both $H$ and $K$ are symmetric groups of sets of $4$ elements (say $X$ and $Y$, respectively), and hence they are isomorphic under the map $\beta_{|X}\mapsto f\beta_{|X} f^{-1}$, for whatever bijection $f\colon X\to Y$.