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Commuting elements in a conjugacy class of finite groups. (English. Russian original) Zbl 1358.20011

Russ. Math. 60, No. 8, 9-16 (2016); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 2016, No. 8, 12-20 (2016).
The authors consider the following conjecture: A nonidentity conjugacy class in a finite simple group contains at least two commuting elements. In the given paper, they verify this conjecture for any conjugacy class of involutions, for the projective groups \(L_n(q)\), the alternating groups \(A_n\) with \(n\geq 5\) and the sporadic groups. The proofs use the theory of left distributive groupoids (see surveys [Russ. Math. Surv. 39, No. 6, 211–212 (1984); translation from Usp. Mat. Nauk 39, No. 6(240), 191–192 (1984; Zbl 0581.20067); “Quasigroups”, Itogi Nauki Tekhn. Algebra. Topology. Geometry Uspekhi Mat. Nauk 89, No. 6, 191–198 (1984)] of the first author and the paper [J. Math. Sci., New York 130, No. 3, 4720–4723 (2005; Zbl 1144.20321); translation from Zap. Nauchn. Semin. POMI 305, 136–143 (2003)] of the second author), [J. H. Conway et al., Atlas of finite groups. Maximal subgroups and ordinary characters for simple groups. Oxford: Clarendon Press (1985; Zbl 0568.20001)] and some properties of the natural \(\mathrm{GL}_n(q)\)-module.

MSC:

20D06 Simple groups: alternating groups and groups of Lie type
20D08 Simple groups: sporadic groups
20E45 Conjugacy classes for groups
20N02 Sets with a single binary operation (groupoids)
20N05 Loops, quasigroups
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References:

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