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 ?