Nilpotência de grupos e álgebras de Lie admitindo grupos de Frobenius de automorfismos
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2021-04-07
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Universidade Federal de Goiás
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Let A be a group acting on a group G. It is known that some properties of G are influenced by CG(A), for example, suppose that a Frobenius group FH acts on a finite group G, we known that if CG(F) = 1 and CG(H) is nilpotent, then G is nilpotent and, adding the hypothesis that F is cyclic, we have that the nilpotency class of G is bounded in terms of the order of H and the nilpotency class of CG(H). Until now, it was not evident, considering the hypotheses mentioned above, if the nilpotency class of G could be made independent of the order of H. In this dissertation, we show that exists a family G of finite nilpotent groups, of unbounded nilpotency class, such that each group in G admits a metacyclic Frobenius group of automorphisms so that CG(F) = 1 and CG(H) is abelian, thus evidencing the essential dependency of the order of H in the nilpotency class of G.
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DAL BERTO, L. M. L. Nilpotência de grupos e álgebras de Lie admitindo grupos de Frobenius de automorfismos. 2021. 102 f. Dissertação (Mestrado em Matemática) - Universidade Federal de Goiás, Goiânia, 2021.