Open AccessFrobenius groups as groups of automorphisms
Author(s) -
N. Yu. Makarenko,
Pavel Shumyatsky
Publication year - 2010
Publication title -
proceedings of the american mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.968
H-Index - 84
eISSN - 1088-6826
pISSN - 0002-9939
DOI - 10.1090/s0002-9939-2010-10494-x
Subject(s) - algorithm , annotation , computer science , artificial intelligence
We show that if G F H GFH is a double Frobenius group with “upper” complement H H of order q q such that C G ( H ) C_G(H) is nilpotent of class c c , then G G is nilpotent of ( c , q ) (c,q) -bounded class. This solves a problem posed by Mazurov in the Kourovka Notebook . The proof is based on an analogous result on Lie rings: if a finite Frobenius group F H FH with kernel F F of prime order and complement H H of order q q acts on a Lie ring K K in such a way that C K ( F ) = 0 C_K(F)=0 and C K ( H ) C_K(H) is nilpotent of class c c , then K K is nilpotent of ( c , q ) (c,q) -bounded class.