Centers of Sylow subgroups and automorphisms
- PDF / 226,212 Bytes
- 14 Pages / 429.408 x 636.768 pts Page_size
- 80 Downloads / 198 Views
CENTERS OF SYLOW SUBGROUPS AND AUTOMORPHISMS BY
George Glauberman∗ Department of Mathematics, University of Chicago 5734 S. University Ave, Chicago, IL 60637, USA e-mail: [email protected] AND
Robert Guralnick∗∗ Department of Mathematics, University of Southern California Los Angeles, CA 90089-2532, USA e-mail: [email protected] AND
Justin Lynd† Department of Mathematics, University of Louisiana at Lafayette Maxim Doucet Hall, Lafayette, LA 70504, USA e-mail: [email protected] AND
Gabriel Navarro†† Departament of Mathematics, Universitat de Val`encia 46100 Burjassot, Val`encia, Spain e-mail: [email protected]
∗ The first author thanks the Simons Foundation very much for its support by a
Collaboration grant.
∗∗ The second author gratefully acknowledges the support of the NSF grants DMS-
1600056 and DMS-1901595.
† The third author gratefully acknowledges support from NSF Grant DMS-1902152. †† The research of the fourth author is supported by Ministerio de Ciencia e Inno-
vaci´ on PID2019-103854GB-I00 and FEDER funds. Received June 29, 2019 and in revised form September 16, 2019
1
2
G. GLAUBERMAN ET AL.
Isr. J. Math.
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.
1. Introduction Let p be a prime. There has been quite a lot of interest in the problem of characterizing automorphisms of order a power of p centralizing a Sylow psubgroup of a finite group G. In particular, Question 14.1 of the Kourovka Notebook [14] which was posed in 1999 asked whether if G has no non-trivial normal odd order subgroups (i.e., O2 (G) = 1), then for any such automorphism a with p = 2, a2 is inner. This had already been answered in the affirmative in Glauberman [4, Corollary 8] in 1968. By taking G = An with n ≥ 6 with n ≡ 2, 3 mod 4 and g ∈ Sn a transposition, one sees that it is not always true that a is inner. If p is odd, then under the assumptions that Op (G) = Op (G) = 1, Gross [10] showed that any such automorphism is inner. Gross used the classification of finite simple groups while Glauberman did not. In this note, we show how to extend the result of Gross allowing the possibility of nontrivial Op (G) (and using the classification of finite simple groups). This was conjectured by Gross in [10] and a partial result was obtained by Murai [16]. We complete the proof and show: Theorem 1.1: Let p be an odd prime and L a finite group with Op (L) = 1. Suppose that a is an automorphism of L whose order is a power of p and a centralizes a Sylow p-subgroup of L. Then a is an inner automorphism of L. It is not hard to see that Theorem 1.1 can fail if Op (L) = 1. For example, take L = G×Op (L) and choose a to be a non-inner automorphism of Op (L), viewed as an automorphism of L acting trivially on G. A consequence of Theorem 1.1 is the following. Recall that F ∗ (G) is the generalized Fitting subgroup of G. Corollary 1.2: Let p be an odd prime, and let G
Data Loading...
