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

If all distinct Sylow 2-subgroups met trivially, part (b) would give $n_2\equiv 1(\mathrm{mod}8)$. But Sylow gives $n_2\mid 21$ and $n_2$ odd, so $n_2\in \{3,7,21\}$ after excluding normality, none congruent to one modulo eight. Thus some distinct pair has nontrivial intersection. That intersection is a nontrivial finite 2-group, so Cauchy's theorem supplies an element of order two.

Solved by gpt-5.6-sol high.