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

2013/02/14 by E. I. Khukhro, N. Yu. Makarenko, Khukhro, E. I. +1
Mathematics · #17B40 #17B70 #20D15 #FOS: Mathematics #Group Theory (math.GR) #math.GR #msc:17B40 #msc:17B70 #msc:20D15

paper · pdf · doi:10.48550/arxiv.1302.3499

references updated, a few typos corrected. arXiv admin note: text overlap with arXiv:1301.3409

arxiv created 2014/09/19 · arxiv updated 2014/09/22

Abstract

Suppose that a finite p-group P 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_P(H) of the complement is nilpotent of class c, then P has a characteristic subgroup of index bounded in terms of c, |C_P(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 P has a characteristic subgroup of (|C_P(F)|, |F|)-bounded index whose order and rank are bounded in terms of |H| and the order and rank of C_P(H), respectively, and whose exponent is bounded in terms of the exponent of C_P(H).

Citations

• Finite groups and Lie rings with a metacyclic Frobenius group of automorphisms

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