Let H = {β ∈ S5 | β(1) = 1} and K = {β ∈ S5 | β(2) = 2}. Prove that H is isomorphic to K

Question

Let H = {β ∈ S5 | β(1) = 1} and K = {β ∈ S5 | β(2) = 2}. Prove that H is isomorphic to K.

This problem has been posted on here before but the only response to it is the hint:

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

I still don't understand it. Does this mean β(1 2) = (1 2)?

Answer 1

Instead of writing down an isomorphism between $H$ and $K$, one also can just rewrite $K$ by renaming $1$ into $2$ and vice versa. So $S_5$ in the second case is the group of all bijections from $\{2,1,3,4,5\}$ to itself. Then simply $H=K$.

Answer 2

Hint:

$$\beta_K = (12)\beta_H(12)$$

Each $\beta_K \in K$ can be written that way with a $\beta_H \in H$ and vice versa.