20.48 (2022)

Suppose that $P$ is a finite $p$-group with a non-trivial partition (which is equivalent to having proper Hughes subgroup $H_p(P) := \langle g \in P \mid g^p \neq 1 \rangle \neq P$). If $P$ admits a fixed-point-free automorphism of prime order, must $P$ be of exponent $p$?