Abstract

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 CG(H) of the complement is nilpotent of class c,
then G has a characteristic subgroup of index bounded in terms of c, jCG(F)j, and
jFj whose nilpotency class is bounded in terms of c and jHj 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 (jCG(F)j; jFj)-bounded index whose order and rank are
bounded in terms of jHj and the order and rank of CG(H), respectively, and whose
exponent is bounded in terms of the exponent of CG(H).