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We summarize several results about non-simplicity, solvability and normal structure of finite groups related to the number of conjugacy classes appearing in the product or the power of conjugacy classes. We also collect some problems that have only been partially solved.

[EN] We survey known results concerning how the conjugacy classes contained in a normal subgroup and their sizes exert an influence on the normal structure of a finite group. The approach is mainly presented in the framework of graphs associated to the conjugacy classes, which have been introduced and developed in the past few years. We will see how the properties of these graphs, along with some extensions of the classic Landau's Theorem on conjugacy classes for normal subgroups, have been used in order to classify groups and normal subgroups satisfying certain conjugacy class numerical conditions. ; Some of the results of this paper are part of the third author's Ph.D. thesis at the University Jaume I of Castellón, who is financially supported by a predoctoral grant of this university. The first and second authors are supported by the Valencian Government, Proyecto PROMETEOII/2015/011. The first and the third authors are also partially supported by Universitat Jaume I, grant P11B-2015-77. ; Beltrán, A.; Felipe, MJ.; Melchor, C. (2018). Conjugacy classes contained in normal subgroups: an overview. International Journal of Group Theory. 7(1):23-26. https://doi.org/10.22108/IJGT.2017.21216 ; S ; 23 ; 26 ; 7 ; 1

[EN] We summarize several results about non-simplicity, solvability and normal structure of finite groups related to the number of conjugacy classes appearing in the product or the power of conjugacy classes. We also collect some problems that have only been partially solved. ; Some results of this survey were obtained during the stay of C. Melchor at the University of Cambridge in autumn 2017, which was financially supported by the grant E-2017-02, Universitat Jaume I of Castellon. She would like to thank R. Camina and the Department of Mathematics for their warm hospitality. A. Beltran and M. J. Felipe are supported by the Valencian Government, Proyecto PROMETEOII/2015/011. ; Beltran, A.; Felipe Román, MJ.; Melchor, C. (2020). Some problems about products of conjugacy classes in finite groups. International Journal of Group Theory. 9(1):59-68. https://doi.org/10.22108/ijgt.2019.111448.1480 ; S ; 59 ; 68 ; 9 ; 1

[EN] Suppose that G is a finite group and K is a non-trivial conjugacy class of G such that KK (-1) = 1 a D a D (-1) with D a conjugacy class of G. We prove that G is not a non-abelian simple group and we give arithmetical conditions on the class sizes determining the solvability and the structure of aOE (c) K > and aOE (c) D >. ; The authors gratefully acknowledge all helpful comments made by the referee. The results in this paper are part of the third author's Ph.D. thesis, and she acknowledges the predoctoral grant PRE-DOC/2015/46, Universitat Jaume I. The first and second authors are supported by the Valencian Government, Proyecto PROMETEOII/2015/011. The first and third authors are also partially supported by Universitat Jaume I, grant P11B2015-77. ; Beltrán, A.; Felipe Román, MJ.; Melchor, C. (2018). Multiplying a conjugacy class by its inverse in a finite group. Israel Journal of Mathematics. 227(2):811-825. https://doi.org/10.1007/s11856-018-1742-9 ; S ; 811 ; 825 ; 227 ; 2 ; E. Adan-Bante, Products of characters with few irreducible constituents, Journal of Algebra 311 (2007), 38–68. ; E. Adan-Bante, Symmetric groups and conjugacy classes, Journal of Group Theory 3 (2008), 371–379. ; Z. Arad and E. Fisman, An analogy between products of two conjugacy classes and products of two irreducible characters in finite groups, Proceedings of the Edinburgh Mathematical Society 30 (1987), 7–22. ; Z. Arad and M. Herzog, Products of Conjugacy Classes in Groups, Lecture Notes in Mathematics, Vol. 1112, Springer-Verlag, Berlin, 1985. ; A. Beltrán, M. J. Felipe and C. Melchor, Squares of real conjugacy classes in finite groups, Annali di Matematica Pura ed Applicata 197 (2018), 317–328. ; R. W. Carter, Finite Groups of Lie Type, Pure and Applied Mathematics (New York), John Wiley & Sons, New York, 1985. ; The GAP Group, GAP - Groups, Algorithms and Programming, Vers. 4.7.7, 2015, https://doi.org/www.gap-system.org ; G. Glauberman, Central elements in core-free groups, Journal of Algebra 4 (1966), 403–420. ...

[EN] Let G be a finite group and N a normal subgroup of G. We determine the structure of N when the graph G(N), which is the graph associated to the conjugacy classes of G contained in N, has no triangles and when the graph consists in exactly one triangle. ; The research of the first and second authors is supported by the Valencian Government, Proyecto PROMETEOII/2015/011. The first and the third authors are also partially supported by Universitat Jaume I, Grant P11B2015-77. ; Beltrán, A.; Felipe Román, MJ.; Melchor, C. (2017). Triangles in the graph of conjugacy classes of normal subgroups. Monatshefte für Mathematik. 182(1):5-21. https://doi.org/10.1007/S00605-015-0866-9 ; S ; 5 ; 21 ; 182 ; 1 ; Bertram, E.A., Herzog, M., Mann, A.: On a graph related to conjugacy classes of groups. Bull. London Math. Soc. 22(6), 569-575 (1990) ; Beltrán, A., Felipe, M.J., Melchor, C.: Graphs associated to conjugacy classes of normal subgroups in finite groups. J. Algebra 443, 335-348 (2015) ; Camina, A.R.: Arithmetical conditions on the conjugacy class numbers of a finite group. J. London Math. Soc. 2(5), 127-132 (1972) ; Deaconescu, M.: Classification of finite groups with all elements of prime order. Proc. Am. Math. Soc. 106(3), 625-629 (1989) ; Doerk, K., Hawkes, T.: Finite soluble groups. de Gruyter Expositions in Mathematics, vol. 4. Walter de Gruyter, Berlin (1992) ; Fang, M., Zhang, P.: Finite groups with graphs containing no triangles. J. Algebra 264(2), 613-619 (2003) ; Higman, G.: Finite groups in which every element has prime power order. J. London Math. Soc. 32, 335-342 (1957) ; Manz, O., Wolf, T.R.: Representations of solvable groups. Cambridge Univ. Press, Cambridge (1993) ; Riese, U., Shahabi, M.A.: Subgroups which are the union of four conjugacy classes. Commun. Algebra 29(2), 695-701 (2001) ; Shahryari, M., Shahabi, M.A.: Subgroups which are the union of three conjugate classes. J. Algebra 207(1), 326-332 (1998) ; The GAP Group.: GAP–groups, algorithms and programming, Vers. 4.4.12. (2008). ...

Landau's theorem on conjugacy classes asserts that there are only finitely many finite groups, up to isomorphism, with exactly k conjugacy classes for any positive integer k. We show that, for any positive integers n and s, there exist finitely many finite groups G, up to isomorphism, having a normal subgroup N of index n which contains exactly s non-central G-conjugacy classes. Upper bounds for the orders of G and N are obtained; we use these bounds to classify all finite groups with normal subgroups having a small index and few G-classes. We also study the related problems when we consider only the set of G-classes of prime-power order elements contained in a normal subgroup. ; The results in this paper are part of the third author's Ph.D. thesis at the Jaume I University of Castellon, who is financially supported by a predoctoral grant of the Jaume I University. The research of the first and second authors is supported by the Valencian Government, Proyecto PROMETEOII/2015/011. The first and the third authors are also partially supported by the Jaume I University, grant P11B2015-77. ; Beltran, A.; Felipe Román, MJ.; Melchor, C. (2016). Landau's theorem on conjugacy classes for normal subgroups. International Journal of Algebra and Computation. 26(7):1453-1466. https://doi.org/10.1142/S0218196716500624 ; S ; 1453 ; 1466 ; 26 ; 7

Let G be a finite group and let N be a normal subgroup of G. We determine the structure of N when the diameter of the graph associated to the G-conjugacy classes contained in N is as large as possible, that is, equal to three. ; The research of the first and the second authors is supported by the Valencian Government, Proyecto PROMETEOII/2015/011. The first and the third authors are also partially supported by Universitat Jaume I, grant P11B2015-77. ; Beltran, A.; Felipe Román, MJ.; Melchor, C. (2016). Normal subgroups whose conjugacy class graph has diameter three. Bulletin of the Australian Mathematical Society. 94(2):266-272. https://doi.org/10.1017/S0004972715001860 ; S ; 266 ; 272 ; 94 ; 2

Let G be a finite group and let N be a normal subgroup of G. We attach to N two graphs ΓG(N) and Γ∗ G(N) related to the conjugacy classes of G contained in N and to the set of primes dividing the sizes of these classes, respectively. These graphs are subgraphs of the ordinary ones associated to the conjugacy classes of G, Γ(G) and Γ∗(G), which have been widely studied by several authors. We prove that the number of connected components of both graphs is at most 2, we determine the best upper bounds for the diameters and characterize the structure of N when these graphs are disconnected. ; The research of the first and second authors is supported by the Valencian Government, Proyecto PROMETEOII/2015/011 and by the Spanish Government, Proyecto MTM2010-19938-C03-02. The first author is also supported by Universitat Jaume I, grant P11B2012-05. ; Beltrán, A.; Felipe Román, MJ.; Melchor, C. (2015). Graphs associated to conjugacy classes of normal subgroups in finite groups. Journal of Algebra. 443:335-348. https://doi.org/10.1016/j.jalgebra.2015.06.040 ; S ; 335 ; 348 ; 443

This article analyzes productivity growth for European banks over the 1995–2001 period. In contrast to previous literature, the study encompasses the overwhelming majority of current European Union (EU) countries—all excepting Greece and those joining the EU in 2004. In addition, we use resampling methods so as to gain statistical precision, which turns out to be especially important due to the limitations of the database. In a second stage, additional nonparametric methods—in an attempt to be fully consistent—are used to disentangle some reasons as to why productivity differentials might exist. Results show that productivity growth has occurred in most countries, mainly due to improvement in production possibilities. The bootstrap analysis yields further evidence, as for many firms and countries productivity growth, or decline, is not statistically significant. The two-stage analysis provides some additional insights, suggesting that the relevance of environmental variables found in other studies focusing on efficiency could be lessened when focusing on productivity.