# Help understanding the permutation group action .

I'm trying to understand actions in regards to group theory . specifically in my notes I found the following example :

Say G=$$A_4$$, for $$x \in \{1,2,3,4\}$$, and $$\tau \in A_4$$

We let $$x^{\tau}$$ be the usual permutation action, let $$\tau=(1,4,2,3)$$

Then $$1^{\tau}=4,2^{\tau}=3,3^{\tau}=2, 4^{\tau}=2$$.

My thought's :

I thought this just meant that we perform the permutation on the set

i.e. $$x=\{1,2,3,4\}=\{x_1,x_2,x_3,x_4\}$$

then $$x^{\tau}$$ just permutes these elements according to $$\tau$$.sending $$x_1x_2x_3x_4 \rightarrow x_3x_4x_2x_1$$

giving $$1^{\tau}=4,2^{\tau}=3,3^{\tau}=1,4^{\tau}=2$$

Which almost agrees with the example except for at 3 ? What am I misunderstanding ?

• $3^{\tau}=4^{\tau}$ says that $\tau$ is not injective, a contradiction. – Dietrich Burde Nov 4 '18 at 19:56
• I think $3^\tau=2$ was just a typo, and your $3^\tau=1$ is correct. – Andreas Blass Nov 5 '18 at 1:16