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arXiv · 1901.07048

Centers of Sylow Subgroups and Automorphisms

George Glauberman·Robert Guralnick·Justin Lynd·Gabriel Navarro
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Abstract

Suppose that p is an odd prime and G is a finite group having no normal non-trivial p'-subgroup. We show that if a is an automorphism of G of p-power order centralizing a Sylow p-group of G, then a is inner. This answers a conjecture of Gross. An easy corollary is that if p is an odd prime and P is a Sylow p-subgroup of G, then the center of P is contained in the generalized Fitting subgroup of G. We give two proofs both requiring the classification of finite simple groups. For p=2, the result fails but Glauberman in 1968 proved that the square of a is inner. This answered a problem of Kourovka posed in 1999.

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George Glauberman, Robert Guralnick, Justin Lynd, Gabriel Navarro. 2019-01-21. Centers of Sylow Subgroups and Automorphisms. https://doi.org/10.1007/s11856-020-2064-2

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