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

摘要

Suppose that a finite p-group G admits a Frobenius group of automorphismsudFH with kernel F that is a cyclic p-group and with complement H. It is provedudthat if the fixed-point subgroup CG(H) of the complement is nilpotent of class c,udthen G has a characteristic subgroup of index bounded in terms of c, jCG(F)j, andudjFj whose nilpotency class is bounded in terms of c and jHj only. Examples showudthat the condition of F being cyclic is essential. The proof is based on a Lie ringudmethod and a theorem of the authors and P. Shumyatsky about Lie rings with audmetacyclic Frobenius group of automorphisms FH. It is also proved that G has audcharacteristic subgroup of (jCG(F)j; jFj)-bounded index whose order and rank areudbounded in terms of jHj and the order and rank of CG(H), respectively, and whoseudexponent is bounded in terms of the exponent of CG(H).