Past exam of the mathematics course of the University of Cambridge / 2023 / ib / Paper 2 / 9E / b / ii / Solution

Count pairs $(x,P)$ where $P$ is a Sylow $2$-subgroup fixing $x$. Each of the seven point stabilizers contains three such subgroups, giving 21 pairs. Conversely, a $2$-group acting on seven points has a fixed point, and a Sylow $2$-subgroup cannot fix two points because a two-point stabilizer has order four. Hence every Sylow $2$-subgroup occurs in exactly one pair, so

Similarly, each point stabilizer contains four Sylow $3$-subgroups, giving 28 pairs. A group of order three acting on seven points has a fixed point, and it cannot fix two because a two-point stabilizer is a $2$-group. Thus

The Sylow congruence and divisibility conditions give $n_7\in \{1,8\}$. Regard the faithful action as an embedding $G\le S_7$. If $n_7=1$, its Sylow $7$-subgroup $P$ is normal, so $G\le N_{S_7}(P)$. The stated fact gives $|N_{S_7}(P)|=7\cdot 6=42$, impossible for a subgroup $G$ of order 168. HenceThese are the Sylow counts in a faithful degree-seven action with S4 point stabilizers.

Solved by gpt-5.6-sol high.