# Show that there is an injective homomorphism between $Aut(S_3)$ and $Bij(\{(12), (1 3), (2 3)\}$

Let $$X$$ be equal to $$\{(1 2), (1 3), (2 3)\}$$. How can I prove the existence of an injective homomorphism between $$Aut(S_3)$$ and $$Bij(X)$$ ?

Let $$\varphi$$ be a map between $$Aut(S_3)$$ and $$Bij(X)$$ defined as following :

For $$\psi$$ in $$Aut(S_3)$$, $$\varphi(\psi) = f$$ with $$f : X \longrightarrow X$$ such as $$\forall x \in X$$, $$f(x) = \psi(x)$$

Then I think I can show that $$\varphi$$ is an injective homomorphism.

Do you think it's correct ? Thank you for your help.

• Welcome to MSE. What's the group operation on $X$? Nov 19, 2020 at 9:28
• Surely the bijections of a set of three elements consists of the permutations of these three elements? [Many define a "permutation" in exactly this way.] Nov 19, 2020 at 9:32
• @JoséCarlosSantos $X$ is not a group, but $Bij(X)$ is a group for the composition law. Nov 19, 2020 at 10:08
• @Laury Yes, you got it ! Nov 19, 2020 at 10:21
• @TheSilverDoe Thank you very much ! Nov 19, 2020 at 10:25