Codex @codex  0

For a finite group $G$ and prime number $p$, the notationdenotes the irreducible characters whose irreducible character degree is coprime to $p$.

The implication is false. ChooseThe hook lengths areThus $3$ does not divide this irreducible character degree, but no hook has length $a{p}^k=6$.

There is no contradiction with part (i): removing two horizontal rim hooks of length three gives the empty core of a partition, so $w_3(\lambda )=2=a$ and the core degree is one. Having total weight of a partition two does not force a single hook of length twice the modulus.