Past exam of the mathematics course of the University of Cambridge / 2025 / ib / Paper 1 / 9E / c / i / Solution

Sylow gives $n_7\mid 24$ and $n_7\equiv 1(\mathrm{mod}7)$. Simplicity excludes $n_7=1$, so $n_7=8$. Distinct order-seven subgroups intersect trivially, hence there areelements of order seven.

Solved by gpt-5.6-sol high.