I've formed up a group $G=\{(1),(34),(12)(34),(124)\}$ acting on a set $X= \{1,2,3,4\}$. Knowing the axioms for a group action, that is:
(Compatibility with identity): $e*x=x$ for all $x\in X$
(Compatibility with group operation): $g_1*(g_2*x) = (g_1*g_2)*x,\;\;\forall x\in X,\;\;g_1, g_2\in G$.
I tried checking if the axioms were met. Since, the identity element $(1)$ exists within the group, I supposed that the first axiom was met. For the second axiom I did the following:
Let $b = (34)$ and $c = (12)(34)$, the elements of $G$. For the second axiom to be met $b*(c*x)=(b*c)*x,\;\; x\in X$.
Left Hand Side: $b*(c*x)$, let $x=3$, element of set $X$. Since in $(12)(34)$, $3$ is mapped to $4$, it implies $(c*x)=4$. Then, $b*(4)$ from which I get $3$ again.
Right Hand Side: $(b*c)*x$, let $x=3$, element of $X$. $(b*c)$ gives $(12)$ implying $(12)*x$ from which I get $3$. Hence LHS=RHS.
I thought I proved the group action, but was wrong, the second axiom was not met for other values of $x$. I would really appreciate some help. Is there something wrong with the group I used or is the entire work just wrong?