Khukhro, E. I. and Makarenko, N. Yu.
(2015)
*Finite p-groups with a Frobenius group of automorphisms whose kernel is a cyclic p-group.*
Proceedings of the American Mathematical Society, 143
(5).
pp. 1837-1848.
ISSN 0002-9939

Item Type: | Article |
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Item Status: | Live Archive |
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## 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).

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