20.4 (2022)

A finite group $G$ is said to be cut (or inverse semi-rational) if $\langle x \rangle = \langle y \rangle$ implies that $x$ is conjugate to $y$ or to $y^{-1}$ for all $x, y \in G$.
$\qquad$ a) Let $\mathbb{Q}(G)$ denote the field extension of the rationals obtained by adjoining all entries of the ordinary character table of $G$. Is there $c > 0$ such that $|\mathbb{Q}(G) : \mathbb{Q}| \leqslant c$ for all cut groups? This is true if one assumes in addition that $G$ is solvable (J. F. Tent, J. Algebra, 363 (2012), 73–82).
$\qquad$ b) Is a Sylow 3-subgroup of a cut group also a cut group?
$\qquad$ c) Let $O_p(G)$ denote the largest normal $p$-subgroup of $G$. Let $G$ be a solvable cut group. Is it true that for $p \in \{5, 7\}$ the exponent of $O_p(G)$ divides $p$?

For additional information and some positive results see (Adv. Group Theory Appl., 8B, 2020, 157–160 or https://arxiv.org/abs/2001.02637).