Conjugacy classes of finite groups and graph regularity

Given a finite group $G$, denote by $\Gamma(G)$ the simple undirected graph whose vertices are the distinct sizes of noncentral conjugacy classes of $G$, and set two vertices of $\Gamma(G)$ to be adjacent if and only if they are not coprime numbers. In this note we prove that, if $\Gamma(G)$ is a $k$-regular graph with $k\geq 1$, then $\Gamma(G)$ is a complete graph with $k+1$ vertices

On vanishing class sizes in finite groups

© 2017 Elsevier Inc. Let G be a finite group. An element g of G is called a vanishing element if there exists an irreducible character χ of G such that χ(g)=0; in this case, we say that the conjugacy class of g is a vanishing conjugacy class. In this paper, we discuss some arithmetical properties concerning the sizes of the vanishing conjugacy classes in a finite group

On solvable groups with one vanishing class size

Copyright © The Author(s), 2020. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh. Let G be a finite group, and let cs(G) be the set of conjugacy class sizes of G. Recalling that an element g of G is called a vanishing element if there exists an irreducible character of G taking the value 0 on g, we consider one particular subset of cs(G), namely, the set vcs(G) whose elements are the conjugacy class sizes of the vanishing elements of G. Motivated by the results inBianchi et al. (2020, J. Group Theory, 23, 79-83), we describe the class of the finite groups G such that vcs(G) consists of a single element under the assumption that G is supersolvable or G has a normal Sylow 2-subgroup (in particular, groups of odd order are covered). As a particular case, we also get a characterization of finite groups having a single vanishing conjugacy class size which is either a prime power or square-free

Prime power indices in factorised groups

[EN] Let the group G = AB be the product of the subgroups A and B. We determine some structural properties of G when the p-elements in A. B have prime power indices in G, for some prime p. More generally, we also consider the case that all prime power order elements in A. B have prime power indices in G. In particular, when G = A = B, we obtain as a consequence some known results.The first author is supported by Proyecto Prometeo II/2015/011, Generalitat Valenciana (Spain), and the second author is supported by Proyecto MTM2014-54707-C3-1-P, Ministerio de Economia, Industria y Competitividad (Spain). The results in this paper are part of the third author's Ph.D. thesis, and he acknowledges the predoctoral grant ACIF/2016/170, Generalitat Valenciana (Spain).Felipe Román, MJ.; Martínez-Pastor, A.; Ortiz-Sotomayor, VM. (2017). Prime power indices in factorised groups. Mediterranean Journal of Mathematics. 14(6):1-15. https://doi.org/10.1007/s00009-017-1023-6S115146Amberg, B., Franciosi, S., de Giovanni, F.: Products of Groups. Oxford University Press Inc., New York (1992)Baer, R.: Group elements of prime power index. Trans. Am. Math. Soc. 75, 20–47 (1953)Ballester-Bolinches, A., Cossey, J., Li, Y.: Mutually permutable products and conjugacy classes. Monatsh. Math. 170, 305–310 (2013)Ballester-Bolinches, A., Esteban-Romero, R., Asaad, M.: Products of finite groups, vol. 53 of de Gruyter Expositions in Mathematics, Berlin (2010)Berkovich, Y., Kazarin, L.S.: Indices of elements and normal structure of finite groups. J. Algebra 283, 564–583 (2005)Camina, A.R., Camina, R.D.: Implications of conjugacy class size. J. Group Theory 1(3), 257–269 (1998)Camina, A.R., Shumyatsky, P., Sica, C.: On elements of prime-power index in finite groups. J. Algebra 323, 522–525 (2010)Chillag, D., Herzog, M.: On the length of the conjugacy classes of finite groups. J. Algebra 131, 110–125 (1990)Doerk, K., Hawkes, T.: Finite Soluble Groups, vol. 4 of de Gruyter Expositions in Mathematics, Berlin (1992)Felipe, M.J., Martínez-Pastor, A., Ortiz-Sotomayor, V.M.: On finite groups with square-free conjugacy class sizes. Int. J. Group Theory (to appear)Kurzweil, H., Stellmacher, B.: The theory of finite groups: an introduction. Springer, New York (2004)Liu, X., Wang, Y., Wei, H.: Notes on the length of conjugacy classes of finite groups. J. Pure Appl. Algebra 196, 111–117 (2005

Triangles in the graph of conjugacy classes of normal subgroups

[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/S00605-015-0866-9S5211821Bertram, 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). http://www.gap-system.or

Body composition and somatotype in professional men's handball according to playing positions

Se realizó un estudio descriptivo transversal en 19 jugadores profesionales del Club Balonmano Valladolid. Las mediciones antropométricas fueron realizadas según el protocolo estándar. Se estimaron la masa grasa y ósea, se calculó el somatotipo y se analizaron las diferencias entre las variables en función de la posición. Como resultados, se obtuvo que los pivotes fueron los jugadores más pesados (con mayor porcentaje de masa grasa); los extremos, los más ligeros y los laterales, junto con los pivotes, los más altos. No se observaron diferencias en el IMC en los grupos. En la somatocarta los centrales y laterales se situaron en la zona central; los extremos y los pivotes en la endomorfa-mesomorfa y los porteros en la ecto-endomorfa. Así se evidenció que las variables antropométricas, los datos de composición corporal y la somatocarta de los deportistas confirman las características morfológicas básicas de los jugadores para la posición para la que son más aptosA cross-sectional descriptive study was accomplished in 19 professional players from Valladolid Handball Club. Anthropometric measurements were performed according to standard protocol. Body fat and bone mass were estimated, and the somatotype was calculated. As results, the line players were significantly the heaviest players; the wings were lightest and the backs, with the line players, the tallest. Nevertheless, no significant differences in BMI were observed. Regarding the body composition, the line players showed the highest values of fat-mass. No differences in BMI were observed in the groups. With respect to the somatochart, the center backs and backs were in the central area; wings and line players showed an endomorph-mesomorph development, and goalkeepers were in the ectoendomorph area. As conclusions, anthropometric variables, body composition data and the somatochart of the athletes evaluated confirm the basic morphological characteristics of the players for the position for which they are best suite

A Concerted Kinase Interplay Identifies PPARγ as a Molecular Target of Ghrelin Signaling in Macrophages

The peroxisome proliferator-activator receptor PPARγ plays an essential role in vascular biology, modulating macrophage function and atherosclerosis progression. Recently, we have described the beneficial effect of combined activation of the ghrelin/GHS-R1a receptor and the scavenger receptor CD36 to induce macrophage cholesterol release through transcriptional activation of PPARγ. Although the interplay between CD36 and PPARγ in atherogenesis is well recognized, the contribution of the ghrelin receptor to regulate PPARγ remains unknown. Here, we demonstrate that ghrelin triggers PPARγ activation through a concerted signaling cascade involving Erk1/2 and Akt kinases, resulting in enhanced expression of downstream effectors LXRα and ABC sterol transporters in human macrophages. These effects were associated with enhanced PPARγ phosphorylation independently of the inhibitory conserved serine-84. Src tyrosine kinase Fyn was identified as being recruited to GHS-R1a in response to ghrelin, but failure of activated Fyn to enhance PPARγ Ser-84 specific phosphorylation relied on the concomitant recruitment of docking protein Dok-1, which prevented optimal activation of the Erk1/2 pathway. Also, substitution of Ser-84 preserved the ghrelin-induced PPARγ activity and responsiveness to Src inhibition, supporting a mechanism independent of Ser-84 in PPARγ response to ghrelin. Consistent with this, we found that ghrelin promoted the PI3-K/Akt pathway in a Gαq-dependent manner, resulting in Akt recruitment to PPARγ, enhanced PPARγ phosphorylation and activation independently of Ser-84, and increased expression of LXRα and ABCA1/G1. Collectively, these results illustrate a complex interplay involving Fyn/Dok-1/Erk and Gαq/PI3-K/Akt pathways to transduce in a concerted manner responsiveness of PPARγ to ghrelin in macrophages

Transitive projective planes

A long-standing conjecture is that any transitive finite projective plane is Desarguesian. We make a contribution towards a proof of this conjecture by showing that a group acting transitively on the the points of a non-Desarguesian projective plane must not contain any components