Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 160 / 3 / b

True. A hook length divisible by $ef$ is divisible by $e$. If $\lambda$ is its own Core of a partition $C_e(\lambda )$, it has no such hook, so it is also its own $ef$-core.

False. Let $\lambda =(3)$, $e=2$, and $f=3$. ThenThus core operations for different moduli need not commute.

True. On an $f$-runner partition abacus, taking the $e$-quotient of each runner refines its positions by their residue modulo $e$. The resulting $ef$ runners are indexed by pairs of residues. Ordering those pairs may differ from the conventional residue order modulo $ef$, but only by a permutation. Hence $Q_e(Q_f(\lambda ))$ is a permutation of $Q_{ef}(\lambda )$.

True. Slide beads upward separately in each of the refined residue classes modulo $ef$. Grouping the refined runners first by residue modulo $f$ and then packing each component modulo $e$ produces $C_e(Q_f(\lambda ))$. Packing modulo $ef$ before forgetting the refinement produces $Q_f(C_{ef}(\lambda ))$. These are the same runner configurations, so