Kourovka Notebook Problem 16.60

Definitions and mathematical statement

Let $G$ be a finite group, and let $k(G)$ denote its number of conjugacy classes. Choose one representative $\rho_i\colon G\to\operatorname{GL}(V_i)$ from each equivalence class of irreducible complex representations of $G$, indexed by $1\leq i\leq k(G)$, and set \[ d_i=\dim_{{\mathbb{C}}}V_i,\qquad T(G)=\sum_{i=1}^{k(G)}d_i. \] For an automorphism $\alpha\in\operatorname{Aut}(G)$, define \[ S_{\alpha}=\{g\in G:\alpha(g)=g^{-1}\}. \] The question proposed by D. MacHale is whether \[ \forall\,G\text{ finite group}\quad\forall\,\alpha\in\operatorname{Aut}(G),\qquad T(G)\geq |S_{\alpha}|. \] There is no restriction on the order of $\alpha$ in the supplied statement.

Short English statement

For every finite group and every automorphism, is the number of elements that the automorphism sends to their inverses at most the sum of the degrees of all inequivalent irreducible complex representations?

Sources

• Local UnsolvedMath registry, permanent ID $701252$, exact source problem code catalogue entry, title Kourovka Notebook Problem 16.60. The supplied record labels the problem open; this label is not an independently verified literature assessment. The record was updated at 2026-10-16T14:00:00Z.
• D. MacHale, proposer, 2006. The Kourovka Notebook (Unsolved Problems in Group Theory), Problem 16.60, 16th issue (2006); the supplied source also references the 21st issue edition (2026), arXiv:1401.0300v48. The supplied collection description names E. I. Khukhro and V. D. Mazurov as editors. These bibliographic details are supplied attribution and have not been independently verified.