Can a normal subgroup of a transitive group also be transitive?

[Euge]

Here is this week's POTW:

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Let $p$ be a prime, $G$ a transitive subgroup of the symmetric group $S_p$, and $A$ a nontrivial normal subgroup of $G$. Prove that $A$ is a transitive subgroup of $S_p$.

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[Euge]

This week's problem was correctly solved by johng. Here is his solution:

Spoiler