Some problems about products of conjugacy classes in finite groups

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

MAD+. Introducing Misconceptions in the Temporal Analysis of the Mathematical Modelling Process of a Fermi Problem

This work describes how the combination of the mistakes committed by a group of preservice teachers when solving a Fermi problem, with the representation of the temporal analysis of their resolutions, can offer more in-depth information about their conceptual misconceptions regarding mathematical and modelling concepts. The combined representation allows knowing when mistakes occur and provides a powerful tool for instructors to adapt the teaching-learning processes of mathematics at all levels of education. Our study is based on a recent categorisation of students' mistakes, together with the creation of a new representation tool, called MAD+, that can combine all this information. The macroscopic view provided by the MAD+ diagrams gives insight into the context in which the mistakes take place and makes the analysis of the resolution of a Fermi problem more efficient

La Llorona, la Malinche y la mujer chicana de hoy. Cuando ceda el llanto

La Llorona, la Malinche y la mujer chicana de hoy. Cuando ceda el llant

Multiplying a conjugacy class by its inverse in a finite group

[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/s11856-018-1742-9S8118252272E. 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.orgG. Glauberman, Central elements in core-free groups, Journal of Algebra 4 (1966), 403–420.R. M. Guralnick and G. Navarro, Squaring a conjugacy class and cosets of normal subgroups, Proceedings of the American Mathematical Society 144 (2016), 1939–1945.R. M. Guralnick and G. R. Robinson, On extensions on the Baer–Suzuki theorem, Israel Journal of Mathematics 82 (1993), 281–297.B. Huppert, Character Theory of Finite Groups, De Gruyter Expositions inMathematics, Vol. 25, Walter de Gruyter, Berlin, 1998.I. M. Isaacs, Character Theory of Finite Groups, Pure and Applied Mathematics, Vol. 69, Academic Press, New York–London, 1976.G. Malle, Almost irreducible tensor squares, Communications in in Algebra 27 (1999), 1033–1051.G. O. Michler, Theory of Finite Simple Groups, New Mathematical Monographs, Vol. 8, Cambridge University Press, Cambridge, 2006.J. Moori and H. P. Tong-Viet, Products of conjugacy classes in simple groups, Quaestiones Mathematicae 34 (2011), 433–439

Normal subgroups whose conjugacy class graph has diameter three

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. doi:10.1017/S0004972715001860S26627294

Landau's theorem on conjugacy classes for normal subgroups

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. doi:10.1142/S0218196716500624S1453146626

Cuidado cultural de la enfermera al niño hospitalizado con problemas respiratorios proveniente de la sierra andina. Chiclayo, 2017

La presente de investigación titulada “cuidado cultural de la enfermera al niño hospitalizado con problemas respiratorios proveniente de la sierra andina. Chiclayo, 2017. Tuvo como objetivo: Describir y analizar los cuidados culturales de la enfermera(o) al niño hospitalizado con problemas respiratorios proveniente de la sierra andina. El referencial teórico estuvo sustentado por la teoría transcultural de Madeleine Leininger. Investigación cualitativa con diseño de estudio de caso, los sujetos de estudio fueron las enfermeras que laboran en el área de lactantes, preescolares y emergencia del Hospital Regional Docente Las Mercedes. Chiclayo, delimitadas por saturación o redundancia. Como técnica de recolección de datos se utilizó la entrevista semiestructurada a profundidad y como técnica de análisis de datos el análisis de contenido temático, formándose las siguientes categorías: Divergencias en el cuidado enfermero (a) al niño de la sierra andina con problemas respiratorios con sus sub categorías: Resistencias culturales de los padres frente al modelo biologísta. Predominio del cuidado enfermero (a). Separación entre las creencias culturales y exigencias del cuidado. Como segunda categoría: Reflexionando sobre el cuidado cultural de la enfermera, con sus sub categorías: Preservación, negociación y reestructuración de los valores, creencias y prácticas del niño de la sierra andina. Incluyendo al familiar acompañante en el cuidado durante la hospitalización y como última categoría: Interculturalidad enfermera - familiar del niño de la sierra andina. Así mismo durante toda la investigación se tuvo en cuenta, los principios de rigor científico y de la bioética personalista de Sgreccia

The Impact of Direct Subsidies in Spain before and after the CAP'92 Reform

This paper focuses on the changes in farm e±ciency as a tool for policy analysis. The methodology is applied to the introduction of direct payments (DP) and the price sup- port reduction a®ecting large samples of individual farms with joint animal and vegetal production (comparing before and after CAP'92). The case study is justi¯ed for their relevance for Mediterranean forest and grazing land preservation in Spain. Using a non parametric method (DEA) we do not specify the production function of the farms, and can explain the impact of the direct payments on environmental adaptation and e±ciency for animal orientated farms before and after the (DP) introduction in 1992 (CAP'92 reform). Even though the direct payments increased sharply after the CAP'92 reform, in general, the new direct payment system turns out to be insu±cient to o®- set the fact that less environmentally friendly farms remain much more \e±cient," i.e. pro¯table for the farmer. The paper also studies the relationship of e±ciency with other policy relevant factors such as economic size. The results show that after the CAP'92 reform, the subsidy schedule was even more correlated with farm size than before, which we ¯nd to be counterproductive

Productivity growth in European banking

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

Cuidado cultural de la enfermera al niño hospitalizado con problemas respiratorios proveniente de la sierra andina. Chiclayo, 2017

La presente de investigación titulada “cuidado cultural de la enfermera al niño hospitalizado con problemas respiratorios proveniente de la sierra andina. Chiclayo, 2017. Tuvo como objetivo: Describir y analizar los cuidados culturales de la enfermera(o) al niño hospitalizado con problemas respiratorios proveniente de la sierra andina. El referencial teórico estuvo sustentado por la teoría transcultural de Madeleine Leininger. Investigación cualitativa con diseño de estudio de caso, los sujetos de estudio fueron las enfermeras que laboran en el área de lactantes, preescolares y emergencia del Hospital Regional Docente Las Mercedes. Chiclayo, delimitadas por saturación o redundancia. Como técnica de recolección de datos se utilizó la entrevista semiestructurada a profundidad y como técnica de análisis de datos el análisis de contenido temático, formándose las siguientes categorías: Divergencias en el cuidado enfermero (a) al niño de la sierra andina con problemas respiratorios con sus sub categorías: Resistencias culturales de los padres frente al modelo biologísta. Predominio del cuidado enfermero (a). Separación entre las creencias culturales y exigencias del cuidado. Como segunda categoría: Reflexionando sobre el cuidado cultural de la enfermera, con sus sub categorías: Preservación, negociación y reestructuración de los valores, creencias y prácticas del niño de la sierra andina. Incluyendo al familiar acompañante en el cuidado durante la hospitalización y como última categoría: Interculturalidad enfermera - familiar del niño de la sierra andina. Así mismo durante toda la investigación se tuvo en cuenta, los principios de rigor científico y de la bioética personalista de Sgreccia