If we have a group G acting on a Set X define the group homomorphism$$ \varphi:G\to S_x $$ where$$\varphi(g)=\sigma_g ,\sigma_g(x)=g*x=x$$
I am trying to show that if this group homomorphism is surjective then $*$ is transitive. I'd like to see if I have my ideas/understanding clear and if there are points I am missing.
If the group homomorphism is surjective that means for some $g\in G$ we have $\varphi(g)$ that maps every to every permutation in $S_x$
Since $g*x$ is a permutation on $X$ then if we have an $x\in X$, $\varphi(g)$ will send it to itself or another element in X. Since $\varphi$ is surjective there is going to be a permutation that sends a fixed $x$ to every $y\in X$. Which makes $*$ transitive.
Also do we need the group homormorphism to be surjetive for transitivity? If $S_x$ has n elements then there are n! permutations. I believe you need less than that to have this property hold true (atleast for $3\leqslant n$.)