Centers of Sylow Subgroups and Automorphisms
| Autor: | Glauberman, George, Guralnick, Robert, Lynd, Justin, Navarro, Gabriel |
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| Rok vydání: | 2019 |
| Předmět: | |
| Zdroj: | Israel J. Math. 240 (2020), no. 1, 253-266 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1007/s11856-020-2064-2 |
| Popis: | 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. Comment: There was a change of authors from the previous version and a considerable difference in the article |
| Databáze: | arXiv |
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