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Centers of Sylow Subgroups and Automorphisms

Authors :
Glauberman, George
Guralnick, Robert
Lynd, Justin
Navarro, Gabriel
Source :
Israel J. Math. 240 (2020), no. 1, 253-266
Publication Year :
2019

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.<br />Comment: There was a change of authors from the previous version and a considerable difference in the article

Details

Database :
arXiv
Journal :
Israel J. Math. 240 (2020), no. 1, 253-266
Publication Type :
Report
Accession number :
edsarx.1901.07048
Document Type :
Working Paper
Full Text :
https://doi.org/10.1007/s11856-020-2064-2