Baer–Suzuki theorem

In mathematical finite group theory, the Baer–Suzuki theorem, proved by Baer (1957) and Suzuki (1965), states that if any two elements of a conjugacy class C of a finite group generate a nilpotent subgroup, then all elements of the conjugacy class C are contained in a nilpotent subgroup. Alperin & Lyons (1971) gave a short elementary proof.
References

Alperin, J. L.; Lyons, Richard (1971), "On conjugacy classes of p-elements", Journal of Algebra 19: 536–537, doi:10.1016/0021-8693(71)90086-x, ISSN 0021-8693, MR 0289622
Baer, Reinhold (1957), "Engelsche Elemente Noetherscher Gruppen", Mathematische Annalen 133: 256–270, doi:10.1007/BF02547953, ISSN 0025-5831, MR 0086815
Gorenstein, D. (1980), Finite groups (2nd ed.), New York: Chelsea Publishing Co., ISBN 978-0-8284-0301-6, MR 569209
Suzuki, Michio (1965), "Finite groups in which the centralizer of any element of order 2 is 2-closed", Annals of Mathematics. Second Series 82: 191–212, ISSN 0003-486X, JSTOR 1970569, MR 0183773

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