An elementary approach for a description of the Fitting subgroup and soluble radical of a finite group G
DOI:
https://doi.org/10.35819/remat2021v7i2id5193Keywords:
Group, Nilpotent Groups, Soluble Groups, Fitting Subgroup, Soluble RadicalAbstract
This work presents an approach that prioritizes the use of Isomorphism Theorems of Groups to study soluble groups and nilpotent groups, which aim to describe the soluble radical S(G) of a finite group G as the largest normal solvable subgroup of G and the Fitting subgroup F(G) of a finite group G as the largest normal nilpotent subgroup of a finite group G. As an application, we show that this description allow us to verify whether S(G) and F(G) are examples of a subgroup class defined in Deaconescu and Walls (2011) for which there is a generalization of a classic result that relates a group G with its automorphism group Aut(G).
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BURNSIDE, W. The Theory of Groups of Finite Order. 2. ed. Cambridge: Cambridge University Press, 1911.
CONRAD, B. Solvable and nilpotent groups. Notas de aula. [2011]. Disponível em: http://math.stanford.edu/~conrad/210BPage/handouts/SOLVandNILgroups.pdf. Acesso em: 12 ago. 2020.
CONRAD, K. Subgroup series II. Notas de aula. [entre 2015 e 2020]. Disponível em: https://kconrad.math.uconn.edu/blurbs/grouptheory/subgpseries2.pdf. Acesso em: 14 ago. 2020.
DEACONESCU, M.; WALLS, G. L. On the group of automorphisms of a group. The American Matemathical Monthly, v. 118, n. 5, p. 452-455, maio 2011.
GALLIAN, J. A. Contemporary Abstract Algebra. 8. ed. Boston: Cengage Learning, 2013.
GARCIA, A.; LEQUAIN, Y. Elementos de Álgebra. Rio de Janeiro: IMPA, 2001.
GOMES, J. R. O. Grupos de Automorfismos de Grupos. Orientador: Marcello Fidélis. 2021. 34f. Monografia (Conclusão de Curso de Licenciatura em Matemática) - Departamento de Tecnologias e Linguagens, Instituto Multidisciplinar, Universidade Federal Rural do Rio de Janeiro, Nova Iguaçu, RJ, 2021.
HALL Jr., M. The Theory of Groups. New York: The Macmillan Company, 1959.
ROTMAN, J. J. Galois Theory. New York: Springer-Verlag, 1990.
ZASSENHAUS, H. The Thery of Groups. New York: Chelsea Publishing Company, 1949.
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https://orcid.org/0000-0002-0893-7426
