Suppose that a finite group $G$ admits a Frobenius group of automorphisms
$FH$ with kernel $F$ and complement $H$ such that the fixed-point subgroup of
$F$ is trivial: $C_G(F)=1$. In this situation various properties of $G$ are
shown to be close to the corresponding properties of $C_G(H)$. By using
Clifford's theorem it is proved that the order $|G|$ is bounded in terms of
$|H|$ and $|C_G(H)|$, the rank of $G$ is bounded in terms of $|H|$ and the rank
of $C_G(H)$, and that $G$ is nilpotent if $C_G(H)$ is nilpotent. Lie ring
methods are used for bounding the exponent and the nilpotency class of $G$ in
the case of metacyclic $FH$. The exponent of $G$ is bounded in terms of $|FH|$
and the exponent of $C_G(H)$ by using Lazard's Lie algebra associated with the
Jennings--Zassenhaus filtration and its connection with powerful subgroups. The
nilpotency class of $G$ is bounded in terms of $|H|$ and the nilpotency class
of $C_G(H)$ by considering Lie rings with a finite cyclic grading satisfying a
certain `selective nilpotency' condition. The latter technique also yields
similar results bounding the nilpotency class of Lie rings and algebras with a
metacyclic Frobenius group of automorphisms, with corollaries for connected Lie
groups and torsion-free locally nilpotent groups with such groups of
automorphisms. Examples show that such nilpotency results are no longer true
for non-metacyclic Frobenius groups of automorphisms.Comment: 31 page

Belyaev V. V.

Dade E. C.

Evgeny Khukhro

Hall P.

Khukhro E. I.

Kreknin V. A.

Kuzucuoglu M.

Lazard M.

Mal’cev A. I.

Natalia Makarenko

Pavel Shumyatsky

English

arXiv

Crossref

Forum Mathematicum

Frobenius groups of automorphisms and their fixed points

Suppose that a finite group $G$ admits a Frobenius group of
automorphisms $FH$ with kernel $F$ and complement $H$ such that
the fixed-point subgroup of $F$ is trivial: $C_G(F)=1$. In this
situation various properties of $G$ are shown to be close to the
corresponding properties of $C_G(H)$. By using Clifford's theorem
it is proved that the order $|G|$ is bounded in terms
of $|H|$ and $|C_G(H)|$, the rank of $G$ is bounded in terms
of $|H|$ and the rank of $C_G(H)$, and
that $G$ is nilpotent if $C_G(H)$ is nilpotent. Lie ring methods
are used for bounding the exponent
and the nilpotency class
of $G$ in the case of metacyclic $FH$. The exponent of $G$ is
bounded in terms of $|FH|$ and the exponent of $C_G(H)$ by using Lazard's
Lie algebra associated with the Jennings--Zassenhaus
filtration and its connection with powerful subgroups. The
nilpotency class of $G$ is bounded in terms of $|H|$ and the
nilpotency class of $C_G(H)$ by considering Lie rings with a
finite cyclic grading satisfying a certain `selective nilpotency'
condition. The latter technique also yields similar results
bounding the nilpotency class of Lie rings and algebras with a
metacyclic Frobenius group of automorphisms, with corollaries for
connected Lie groups and torsion-free locally nilpotent groups
with such groups of automorphisms. Examples show that such nilpotency results
are no longer true for non-metacyclic Frobenius groups of automorphisms

Khukhro, E. I.

Makarenko, N. Yu.

Shumyatsky, P.

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Frobenius groups of automorphisms and their fixed points

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