# Avoided a Duplicate Question

## Problem

I encountered the following math problem, so I typed in on Mathematics Stack Exchange. Then, a list of similar posts appeared.

Let $E = {1,\dots,p}$, where $p$ is a prime number. $G$ is a transitive subgroup of $S_p$, and $H$ is a nontrivial normal subgroup of $G$. Show that $H$ acts transitively on $E$.

I notice that $\lvert G \cdot x \rvert = \lvert E \rvert = p$ because $G$ acts transitively on $E$. Now, I try to show the same for $H$, but I am stuck at $\lvert H \cdot x \rvert$. May I say that for all $g \in G$, $\lvert gH \cdot x \rvert = \lvert H \cdot x \rvert$? Why can’t $H$ fix $x$?

Is it a possible duplicate of another question?