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Avoided a Duplicate Question

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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?

Reason to think about this

If the answer is yes, then my post is going to be quickly flagged as duplicate by experienced users on this site, especially those who are strong at abstract algebra, and I’ll lose reputation for that.


I’ve really found the same problem on Mathematics Stack Exchange by luck.

Lessons learnt

I viewed this question four days ago. When I tried to access it again tonight from my web browsing history, I found this a bit hard. Therefore, as a post on Mathematics Meta stack Exchange suggests, Vote early, vote often.