A cute question on group action Let $G$ be a subgroup of $S_n$ that acts transitively on $(1,2,...,n)$. Let $N$ be a non trivial normal subgroup of $G$. Does $N$ act transitively on the set? Its true when $n$ is prime.--prophetes.ai

A cute question on group action Let $G$ be a subgroup of $S_n$ that acts transitively on $(1,2,...,n)$. Let $N$ be a non trivial normal subgroup of $G$. Does $N$ act transitively on the set? Its true when $n$ is prime.

Take any non-simple group $A$, and let it act on itself by left translations. If $A$ has $n$ elements, this allows you to think of $A$ as a subgroup of $S_n$. Now consider any proper non-trivial normal subgroup $B$ of $A$.