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20.74 (2022)

Open

a) Conjecture: there are only finitely many nonabelian finite simple groups $G$ that have a normal subset $S$ closed under inversion such that $|S| > |G|/\log_2 |G|$ and $S^2 \neq G$.

(b) The same conjecture for the groups $\text{PSL}(2, p)$.

Contributor: L. Pyber

Progress

(a) Comment of 2024: This is true for the class of alternating groups (M. Larsen, P. H. Tiep, Preprint, 2023, arXiv:2305.11806) and for any class of groups of Lie type of bounded Lie rank (S. V. Skresanov, Preprint, 2024, https://arxiv.org/abs/2406.12506).

*(b) The conjecture is proved for the groups $\text{PSL}(2, p)$ (S. V. Skresanov, Preprint, 2024, https://arxiv.org/abs/2406.12506).

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