## Abstract

Suppose that a finite group *G* admits a Frobenius group of automorphisms *BC* of coprime order with kernel *B* and complement *C* such that *C*
_{
G
}(*C*) is abelian. It is proved that if *B* is abelian of rank at least two and \({[C_G(u), C_G(v),\dots,C_G(v)]=1}\) for any \({u,v\in B{\setminus}\{1\}}\), where *C*
_{
G
}(*v*) is repeated *k* times, then *G* is nilpotent of class bounded in terms of *k* and |*C*| only. It is also proved that if *B* is abelian of rank at least three and *C*
_{
G
}(*b*) is nilpotent of class at most *c* for every \({b \in B{\setminus}\{1\}}\), then *G* is nilpotent of class bounded in terms of *c* and |*C*|. The proofs are based on results on graded Lie rings with many commuting components.

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## References

- 1
Hall P.: Some sufficient conditions for a group to be nilpotent. Ill. J. Math.

**2**, 787–801 (1958) - 2
Higman G.: Groups and rings which have automorphisms without non-trivial fixed elements. J. Lond. Math. Soc.

**32**, 321–334 (1957) - 3
Khukhro E.I.: Groups and Lie rings admitting an almost regular automorphism of prime order. Mat. Sb.

**181**(9), 1207–1219 (1990) English transl., Math. USSR Sbornik 71(9), 51–63 (1992) - 4
Khukhro E.I.: On the solvability of Lie rings with an automorphism of finite order. Sib. Math. J.

**42**, 996–1000 (2001) - 5
Khukhro E.I.: Finite groups of bounded rank admitting an automorphism of prime order. Sib. Math. J.

**43**, 955–962 (2002) - 6
Khukhro E.I.: Graded Lie rings with many commuting components and an application to 2-Frobenius groups. Bull. Lond. Math. Soc.

**40**, 907–912 (2008) - 7
Khukhro, E.I.: Lie rings with a finite cyclic grading in which there are many commuting components. Sib. Electron. Math. Rep.

**6**, 243–250 (2009) (Russian). http://semr.math.nsc.ru - 8
Khukhro E.I., Makarenko N.Yu., Shumyatsky P.: Nilpotent ideals in graded Lie algebras and almost constant-free derivations. Commun. Algebra

**36**, 1869–1882 (2008) - 9
Khukhro E.I., Shumyatsky P.V.: On fixed points of automorphisms of Lie rings and locally finite groups. Algebra Logic

**34**, 395–405 (1995) - 10
Khukhro E.I., Shumyatsky P.: Lie algebras with almost constant-free derivations. J. Algebra

**306**, 544–551 (2006) - 11
Kreknin V.A.: The solubility of Lie algebras with regular automorphisms of finite period. Dokl. Akad. Nauk SSSR

**150**, 467–469 (1963) English transl., Math. USSR Doklady 4, 683–685 (1963) - 12
Kreknin V.A., Kostrikin A.I.: Lie algebras with regular automorphisms. Dokl. Akad. Nauk SSSR

**149**, 249–251 (1963) English transl., Math. USSR Doklady 4, 355–358 - 13
Makarenko N.Yu.: Graded Lie algebras with few non-trivial components. Sibirsk. Mat. Zh.

**48**, 116–137 (2007) English transl., Sib. Math. J. 48, 95–111 (2007) - 14
Makarenko N.Yu., Khukhro E.I.: Almost solubility of Lie algebras with almost regular automorphisms. J. Algebra

**277**, 370–407 (2004) - 15
Shalev A.: Automorphisms of finite groups of bounded rank. Israel J. Math.

**82**, 395–404 (1993) - 16
Shumyatsky P.: On locally finite groups and the centralizers of automorphisms. Boll. Unione Mat. Italiana

**4**, 731–736 (2001) - 17
Shumyatsky P.: Finite Groups and the Fixed Points of Coprime Automorphisms. Proc. Am. Math. Soc.

**129**, 3479–3484 (2001) - 18
Stewart A.G.R.: On the class of certain nilpotent groups. Proc. R. Soc. Lond. Ser. A

**292**, 374–379 (1966) - 19
Winter D.J.: On groups of automorphisms of Lie algebras. J. Algebra

**8**, 131–142 (1968)

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Communicated by J.S. Wilson.

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Khukhro, E.I., Shumyatsky, P. Nilpotency of finite groups with Frobenius groups of automorphisms.
*Monatsh Math* **163, **461–470 (2011). https://doi.org/10.1007/s00605-010-0198-8

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### Keywords

- Frobenius group of automorphisms
- Centralizer
- Nilpotent
- Graded Lie ring

### Mathematics Subject Classification (2000)

- Primary 20D45
- Secondary 17B70
- 20D15
- 20F40