21.1 (2026)

Let $n$ be a positive integer. For a finite group $K$ and an automorphism $\phi$ of $K$ of order dividing $n$, let $X_{n,\phi}(K) := \{x \in K \mid x x^\phi \dots x^{\phi^{n-1}} = 1\}$. Let $c_n$ be the supremum of the ratios $|X_{n,\phi}(H)|/|H|$ over all finite groups $H$ and their automorphisms $\phi \in \text{Aut}(H)$ such that $\phi^n = \text{id}$ and $X_{n,\phi}(H) \neq H$.
$\qquad$ a) Let $n > 1$ be a positive integer such that $c_d < 1$ for all prime power divisors $d$ of $n$. Is it true that $c_n < 1$?
$\qquad$ b) For a finite group $G$ and a positive integer $n$, the generalized Hughes–Thompson subgroup is defined as $H_n(G) = \langle x \in G \mid x^n \neq 1 \rangle$. Suppose that $n$ is a positive integer for which there is a positive integer $k_n$ depending only on $n$ such that $|G : H_n(G)| \leqslant k_n$ for all finite groups $G$ with $H_n(G) \neq 1$. Is it true that then $c_n < 1$? This question is open even when $n \geqslant 5$ is prime.