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

Abstract

Suppose that a finite p p -group P P admits a Frobenius group of automorphisms F H FH with kernel F F that is a cyclic p p -group and with complement H H . It is proved that if the fixed-point subgroup C P ( H ) C_P(H) of the complement is nilpotent of class c c , then P P has a characteristic subgroup of index bounded in terms of c c , | C P ( F ) | |C_P(F)| , and | F | |F| whose nilpotency class is bounded in terms of c c and | H | |H| only. Examples show that the condition of F 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 F H FH . It is also proved that P P has a characteristic subgroup of ( | C P ( F ) | , | F | ) (|C_P(F)|, |F|) -bounded index whose order and rank are bounded in terms of | H | |H| and the order and rank of C P ( H ) C_P(H) , respectively, and whose exponent is bounded in terms of the exponent of C P ( H ) C_P(H) .