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

Khukhro, E. I. and Makarenko, N. Yu. (2013) Finite $p$-groups with a Frobenius group of automorphisms whose kernel is a cyclic $p$-group. [MIMS Preprint]

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 $C_G(H)$.
Item Type: MIMS Preprint finite $p$-group, Frobenius group, automorphism, nilpotency class, Lie ring MSC 2010, the AMS's Mathematics Subject Classification > 20 Group theory and generalizations Professor Evgeny Khukhro 18 Feb 2013 20 Oct 2017 14:13 http://eprints.maths.manchester.ac.uk/id/eprint/1941