Abstract
The aim of this paper is to investigate the behaviour of prime and semiprime subgroups of groups, and their relation with the existence of abelian normal subgroups. In particular, we study the set Spec(G) of all prime subgroups of a group G endowed with the Zariski topology and, among other examples, we construct an infinite group whose proper normal subgroups are prime and form a descending chain of type ω + 1.
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References
Acosta, L., Rubio, I.M.: Spectral Compactification of a ring. Int. Math. Forum 7(19), 925–934 (2012)
Atiyah, M.F., Macdonald, I.G.: Introduction to Commutative Algebra. Addison-Wesley Publishing Co., Reading (1969)
Baer, R.: Nil-Gruppen. Math. Z. 62, 402–437 (1955)
Belluce, L.P.: Spectral spaces and noncommutative rings. Comm. Algebra 19, 1855–1865 (1991)
Belluce, L.P.: Spectral closure for non-commutative rings. Comm. Algebra 25, 1513–1536 (1997)
Facchini, A., Finocchiaro, C.A., Janelidze, G.: Abstractly constructed prime spectra. Algebra Universalis 83(1), 8–38 (2022)
Facchini, A., Lucchini, A.: The Krull-Schmidt theorem holds for finite direct products of biuniform groups. Algebr. Represent. Theory 21(2), 309–329 (2018)
Gilmer, R.: Commutative rings in which each prime ideal is principal. Math. Ann. 183, 151–158 (1969)
Janelidze, G., Tholen, W.: Characterization of torsion theories in general categories. In: Davydov, A., Batanin, M., Johnson, M., Lack, S., Neeman, A. (eds.) Categories in Algebra, Geometry and Mathematical Physics, pp 249–256. Contemp. Math. 431, Amer. Math. Soc., Providence (2007)
Klep, I., Tressl, M.: The Prime Spectrum and the Extended Prime Spectrum of Noncommutative Rings. Algebr. Represent. Theory 10, 257–270 (2007)
Kurata, Y.: A decomposition of normal subgroups in a group. Osaka Math. J. 1, 201–229 (1964)
Lam, T.Y.: A First Course in Noncommutative Rings, Second Edition Graduate Texts in Math, vol. 131. Springer, New York (2001)
Leinen, F.: Existentially closed locally finite p-groups. J. Algebra 103, 160–183 (1986)
Levitzki, J.: Prime ideals and the lower radical. Amer. J. Math. 73, 25–29 (1951)
McCoy, N.H.: Prime ideals in general rings. Amer. J. Math. 71, 823–833 (1949)
McLain, D.H.: Finiteness conditions in locally soluble groups. J. London Math. Soc. 34, 101–107 (1959)
Robinson, D.J.S.: Finiteness Conditions and Generalized Soluble Groups. Springer, Berlin (1972)
Schenkman, E.: The similarity between the properties of ideals in commutative rings and the properties of normal subgroups of groups. Proc. Amer. Math. Soc. 9, 375–381 (1958)
Shchukin, K.K.: The RI∗-solvable radical of groups. Mat. Sb. (N.S.) 52(94), 1021–1031 (1960)
Steinfeld, O.: Primelemente und Primradikale in gewissen verbandsgeordneten algebraischen Strukturen. Acta. Math. Acad. Sci. Hungar. 19, 243–261 (1968)
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Presented by: Kenneth Goodearl
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The first author is partially supported by Ministero dell’Università e della Ricerca (progetto di ricerca di rilevante interesse nazionale “Categories, Algebras: Ring-Theoretical and Homological Approaches (CARTHA)”), Fondazione Cariverona (research project “Reducing complexity in algebra, logic, combinatorics -REDCOM” within the framework of the programme Ricerca Scientifica di Eccellenza 2018), and the Department of Mathematics “Tullio Levi-Civita” of the University of Padua (Research programme DOR1828909 “Anelli e categorie di moduli”). The second and the third author are supported by GNSAGA (INdAM) and are members of the non-profit association “Advances in Group Theory and Applications” (www.advgrouptheory.com)
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Facchini, A., de Giovanni, F. & Trombetti, M. Spectra of Groups. Algebr Represent Theor 26, 1415–1431 (2023). https://doi.org/10.1007/s10468-022-10138-1
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DOI: https://doi.org/10.1007/s10468-022-10138-1