• DOI:10.1090/S0002-9939-2015-12287-3
• Corpus ID: 53499764

Finite p -groups with a Frobenius group of automorphisms whose kernel is a cyclic p -group

@inproceedings{Khukhro2013FiniteP,
  title={Finite p -groups with a Frobenius group of automorphisms whose kernel is a cyclic p -group},
  author={E. I. Khukhro and N. Yu. Makarenko},
  year={2013}
}

• E. Khukhro, N. Makarenko
• Published 14 February 2013
• Mathematics

Suppose that a finite $p$-group $G$ admits a Frobenius group of automorphisms $FH$ with kernel $F$ that is a cyclic $p$-group and with complement $H$. It is proved that if the fixed-point subgroup $C_G(H)$ of the complement
is nilpotent of class $c$, then $G$ has a characteristic subgroup of index bounded in terms of $c$,
$|C_G(F)|$, and $|F|$ whose nilpotency class is bounded in terms of $c$ and $|H|$ only. Examples show that the condition of $F$ being cyclic is essential.
The proof is based on a Lie ring method and a theorem of the
authors and P.~Shumyatsky about Lie rings with a metacyclic
Frobenius group of automorphisms $FH$. It is also proved that $G$ has a characteristic subgroup of $(|C_G(F)|, |F|)$-bounded index whose order and rank are bounded in terms of $|H|$ and the order and rank of $C_G(H)$, respectively, and whose exponent is bounded in terms of the exponent of $C_G(H)$.