One-prime power hypothesis for conjugacy class sizes

A finite group G satisfies the one-prime power hypothesis for conjugacy class sizes if any two conjugacy class sizes m and n are either equal or have common divisor a prime power. Taeri conjectured that an insoluble group satisfying this condition is isomorphic to S × A where A is abelian and S(Formula Present.) P SL2(q) for q ∈ {4, 8}. We confirm this conjecture

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

Class sizes of prime-power order p'-elements and normal subgroups

We prove an extension of the renowned Itô’s theorem on groups having two class sizes in three different directions at the same time: normal subgroups, p′p′-elements and prime-power order elements. Let NN be a normal subgroup of a finite group GG and let pp be a fixed prime. Suppose that |xG|=1|xG|=1 or mm for every qq-element of NN and for every prime q≠pq≠p. Then, NN has nilpotent pp-complements.We are very grateful to the referee, who provided us a significant simplification of the last step of the proof of the main theorem and for many comments which have contributed to improve the paper. C. G. Shao wants to express his deep gratitude for the warm hospitality he has received in the Departamento de Matematicas of the Universidad Jaume I in Castellon, Spain. This research is supported by the Valencian Government, Proyecto PROMETEO/2011/30, by the Spanish Government, Proyecto MTM2010-19938-C03-02. The third author is supported by the research Project NNSF of China (Grant Nos. 11201401 and 11301218) and University of Jinan Research Funds for Doctors (XBS1335 and XBS1336).Beltrán, A.; Felipe Román, MJ.; Shao, C. (2015). Class sizes of prime-power order p'-elements and normal subgroups. Annali di Matematica Pura ed Applicata. 194(5):1527-1533. https://doi.org/10.1007/s10231-014-0432-4S152715331945Akhlaghi, Z., Beltrán, A., Felipe, M.J.: The influence of $$p$$ p -regular class sizes on normal subgroups. J. Group Theory. 16, 585–593 (2013)Alemany, E., Beltrán, A., Felipe, M.J.: Nilpotency of normal subgroups having two $$G$$ G -class sizes. Proc. Am. Math. Soc. 139, 2663–2669 (2011)Alemany, E., Beltrán, A., Felipe, M.J.: Finite groups with two $$p$$ p -regular conjugacy class lengths II. Bull. Aust. Math. Soc. 797, 419–425 (2009)Beltrán, A., Felipe, M.J.: Normal subgroups and class sizes elements of prime-power order. Proc. Am. Math. Soc. 140, 4105–4109 (2012)Beltrán, A.: Action with nilpotent fixed point subgroup. Arch. Math. (Basel) 69, 177–184 (1997)Camina, A.R.: Finite groups of conjugate rank 2. Nagoya Math. J. 53, 47–57 (1974)Casolo, C., Dolfi, S., Jabara, E.: Finite groups whose noncentral class sizes have the same $$p$$ p -part for some prime $$p$$ p . Isr. J. Math. 192, 197–219 (2012)Huppert, B.: Character Theory of Finite groups, vol. 25. De Gruyter Expositions in Mathemathics, Berlin, New York (1998)Kleidman, P., Liebeck, M.: The Subgroup Structure of The Finite Classical Groups. London Mathematical Society Lecture Note Series, 129. Cambridge University Press, Cambridge (1990)Kurzweil, K., Stellmacher, B.: The Theory of Finite Groups. An Introduction. Springer, New York (2004)The GAP Group, GAP—Groups, Algorithms and Programming, Vers. 4.4.12 (2008). http://www.gap-system.orgVasiliev, A.V., Vdovin, E.P.: An adjacency criterion for the prime graph of a finite simple group. Algebra Logic 44(6), 381–406 (2005

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

Structure of p-complements in groups with three p-regular class sizes

Let G be a finite p-solvable group for some prime p and suppose that the set of p-regular conjugacy class sizes is {1,m,mn} with (m,n)=1 and m coprime to p. We show that m=qb for some prime q and we describe the structure of the p-complements of G.This research is supported by the Spanish Government, Proyecto MTM2010-19938-C03-02 and by the Valencian Government, Proyecto PROMETEO/2011/30. The first author is also supported by grant Fundacio Caixa-Castello P11B2010-47.Beltrán, A.; Felipe Román, MJ. (2013). Structure of p-complements in groups with three p-regular class sizes. Monatshefte für Mathematik. 171(2):157-167. https://doi.org/10.1007/s00605-012-0466-xS1571671712Akhlaghi, Z., Beltrán, A., Felipe, M.J., Khatami, M.: Finite $$p$$ -solvable groups with three $$p$$ -regular class sizes. Proc. Edinb. Math. Soc. (2012). doi: 10.1017/s0013091511000654Alemany, E., Beltrán, A., Felipe, M.J.: Finite groups with two $$p$$ -regular conjugacy class lengths II. Bull. Aust. Math. Soc. 79, 419–425 (2009)Alemany, E., Beltrán, A.: Felipe, M.J: Itô’s theorem on groups with two class sizes revisited. Bull. Aust. Math. Soc. 85, 476–481 (2012)Beltrán, A., Felipe, M.J.: The structure of finite groups with three class sizes. J. Group Theory 12, 539–553 (2009)Beltrán, A., Felipe, M.J.: Certain relations between $$p$$ -regular class sizes and the $$p$$ -structure of $$p$$ -solvable groups. J. Aust. Math. Soc. 77, 387–400 (2004)Beltrán, A., Felipe, M.J.: Prime powers as conjugacy class lengths of $$\pi$$ -elements. Bull. Austral. Math. Soc. 69, 317–325 (2004)Camina, A.R., Camina, R.D.: Implications of conjugacy class size. J. Group Theory 1, 257–269 (1998)Dolfi, S., Jabara, E.: The structure of finite groups of conjugate rank 2. Bull. Lond. Math. Soc. 41, 916926 (2009)Huppert, B.: Character theory of finite groups. De Gruyter Expositions in Mathematics, vol. 25. Walter de Gruyter & Co, Berlin (1998)Itô, N.: On finite groups with given conjugate types. II. Osaka J. Math. 7, 231–251 (1970)Kurzweil, K., Stellmacher, B.: The theory of finite groups. Springer, New York (2004)Schmidt, R.: Subgroup lattices of groups. Walter de Gruyter, Berlin (1994