Past exam of the mathematics course of the University of Cambridge / 2024 / ib / Paper 1 / 9E / a / ii / Solution

A Sylow $2$-subgroup of $S_4$ has order $8$. For example,It is the normalizer of $⟨(1234)⟩$. The six $4$-cycles form three inverse pairs, and a dihedral group of order eight has a unique cyclic subgroup of order four. Hence the three Sylow $2$-subgroups are