Rank and Order of a Finite Group Admitting a Frobenius Group of Automorphisms

被引：0
作者：
E. I. Khukhro
机构：
[1] Sobolev Institute of Mathematics,
来源：
Algebra and Logic | 2013年 / 52卷
关键词：
finite group; Frobenius group; automorphism; rank; order; p-group;
D O I：
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学科分类号：
摘要：
Suppose that a finite group G admits a Frobenius group FH of automorphisms of coprime order with kernel F and complement H. For the case where G is a finite p-group such that G = [G, F], it is proved that the order of G is bounded above in terms of the order of H and the order of the fixed-point subgroup CG(H) of the complement, while the rank of G is bounded above in terms of |H| and the rank of CG(H). Earlier, such results were known under the stronger assumption that the kernel F acts on G fixed-point-freely. As a corollary, for the case where G is an arbitrary finite group with a Frobenius group FH of automorphisms of coprime order with kernel F and complement H, estimates are obtained which are of the form |G| ≤ |CG(F)| · f(|H|, |CG(H)|) for the order, and of the form r(G) ≤r(CG(F)) + g(|H|,r(CG(H))) for the rank, where f and g are some functions of two variables.
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页码：72 / 78
页数：6
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共 20 条
[1]  
Khukhro EI(2008)Graded Lie rings with many commuting components and an application to 2-Frobenius groups Bull. London Math. Soc. 40 907-912
[2]  
Makarenko NY(2010)Frobenius groups as groups of automorphisms Proc. Am. Math. Soc. 138 3425-3436
[3]  
Shumyatsky P(2010)Nilpotent length of a finite group admitting a Frobenius group of automorphisms with fixed-point-free kernel Algebra Logika 49 819-833
[4]  
Khukhro EI(2011)Fixed points of Frobenius groups of automorphisms Dokl. Akad. Nauk 437 20-23
[5]  
Makarenko NY(2011)On the exponent of a finite group with an automorphism group of order twelve J. Alg. 331 482-489
[6]  
Khukhro EI(2011)Positive laws in fixed points of automorphisms of finite groups J. Pure Appl. Alg. 215 2559-2566
[7]  
Shumyatsky P(2012)Fitting height of a finite group with a Frobenius group of automorphisms J. Alg. 366 1-11
[8]  
Shumyatsky P(2012)Automorphisms of finite p-groups admitting a partition Algebra Logika 51 392-411
[9]  
Shumyatsky P(1987)Powerful J. Alg. 105 484-505
[10]  
Khukhro EI(1968)-groups, 1: Finite groups Math. Z. 103 37-39
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