Elements inverted by an automorphism and the sum of character degrees

Show that an automorphism of a finite group inverts at most as many elements as the sum of the irreducible character degrees.

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1	# Elements inverted by an automorphism and the sum of character degrees
2
3	Frobenius and Schur showed that the number of elements $g$ of a finite group with $g^2 = 1$ is at most the sum of the degrees of the group's irreducible complex representations. MacHale asked whether the same bound holds for the elements inverted by any automorphism, not just the identity. A short proof was proposed in October 2026 and has not yet been confirmed by mathematicians.
4
5	## Problem
6
7	Let $G$ be a finite group and let $T(G) = \sum_{\chi} \chi(1)$ be the sum of the degrees of its irreducible complex characters. For an automorphism $\alpha$ of $G$ put
8
9	$$S_\alpha = \{ g \in G : \alpha(g) = g^{-1} \}.$$
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11	Is it true that $\|S_\alpha\| \le T(G)$ for every $\alpha \in \mathrm{Aut}(G)$?
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13	For $\alpha$ the identity, $S_\alpha$ is the set of elements with $g^2 = 1$, and the inequality is the classical consequence of the Frobenius–Schur count of square roots.
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15	## Background
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17	The question was posed by Des MacHale in the 16th issue of the Notebook (2006). It asks whether a classical identity of Frobenius and Schur extends from the identity automorphism to arbitrary automorphisms, through "twisted" Frobenius–Schur indicators. Groups with automorphisms that invert many elements have also been studied for their own sake; for example Bors showed that a finite group with an automorphism inverting a fixed positive proportion of its elements is close to abelian.
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19	## What is known
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21	- Status in the Notebook (marked ♯). In its latest update (October 2026) the Notebook marks this problem with ♯: an artificial-intelligence system has proposed a solution that mathematicians have not yet confirmed, so the problem is to be regarded as unsolved. Readers are invited to confirm the solution or find mistakes.
22	- The proposed solution. The cited note (A. Ferudun, Zenodo, 18 October 2026) answers yes. It writes $\|S_\alpha\| = \sum_\chi \iota_\alpha(\chi)\,\chi(1)$ with twisted indicators $\iota_\alpha(\chi) = \|G\|^{-1} \sum_{g \in G} \chi(g\,\alpha(g))$ and proves $\|\iota_\alpha(\chi)\| \le 1$ from the orthogonality relations; a second proof restricts $\alpha$ to the fixed points of $\alpha^2$. It also claims that equality holds only when $\alpha^2 = 1$ and all twisted indicators equal 1. The note states that AI tools supported the work and that it is unrefereed. It also reports that a 2026 preprint of Yerrapati, Dixit and Shukla states the inequality, with an incomplete proof of one lemma.
23	- Special cases. The inequality was known for involutory automorphisms (via the twisted Frobenius–Schur count), and Yerrapati, Dixit and Shukla (2026) proved it for all automorphisms of dihedral groups and of groups of order $p$, $2p$ and $p^2$.
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25	## What would count as progress
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27	- Verify or refute the proposed solution: check the bound $\|\iota_\alpha(\chi)\| \le 1$ and the identity $\|S_\alpha\| = \sum_\chi \iota_\alpha(\chi)\chi(1)$, which are short enough to check by hand.
28	- Independent computational evidence: check the inequality for every automorphism of every group in a library of small groups (for example in GAP); any counterexample would be decisive.
29	- Classify the cases of equality $\|S_\alpha\| = T(G)$, as claimed in the note.
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31	## References
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33	1. A. Bors, [Finite groups with an automorphism inverting, squaring or cubing a non-negligible fraction of elements](https://arxiv.org/abs/1601.04311), arXiv:1601.04311 (2016).
34	2. V. S. Yerrapati, R. Dixit and A. K. Shukla, [Twisted Frobenius–Schur indicators and character degree sums in dihedral groups](https://arxiv.org/abs/2605.22127), arXiv:2605.22127 (2026).
35	3. V. S. Yerrapati, R. Dixit and A. K. Shukla, [Complex representations of groups and involutions of its automorphisms](https://arxiv.org/abs/2605.23195), arXiv:2605.23195 (2026).
36	4. A. Ferudun, [Elements inverted by an automorphism and the sum of the character degrees](https://doi.org/10.5281/zenodo.23445966), Zenodo (2026), as cited in the Notebook.
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38	## Source
39
40	Imported from [The Kourovka Notebook 16.60](https://kourovkanotebookorg.wordpress.com/wp-content/uploads/2026/10/21tkt-3.pdf#page=106), curated by E. I. Khukhro and V. D. Mazurov. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.