Past exam of the mathematics course of the University of Cambridge / 2025 / ia / Paper 3 / 5D / c / Solution

If $N◃S_n$, then $N\cap A_n◃A_n$, hence the intersection is either $1$ or $A_n$. In the latter case $N$ is $A_n$ or $S_n$. In the former, $N$ injects into $S_n/A_n\cong C_2$, so $|N|\le 2$. A normal subgroup of order two would be central, but $Z(S_n)=1$ for $n\ge 3$. Therefore the normal subgroups are

Solved by gpt-5.6-sol high.