On the Sylow graph of a finite group

The final publication is available at Springer via http://dx.doi.org/10.1007/s11856-011-0138-xLet G be a finite group and Gp be a Sylow p-subgroup of G for a prime p in pi(G), the set of all prime divisors of the order of G. The automiser Ap(G) is defined to be the group NG(Gp)/GpCG(Gp). We define the Sylow graph gamma A(G) of the group G, with set of vertices pi(G), as follows: Two vertices p, q ¿ ¿(G) form an edge of ¿A(G) if either q ¿ ¿(Ap(G)) or p ¿ ¿(Aq(G)). The following result is obtained: Theorem: Let G be a finite almost simple group. Then the graph ¿A(G) is connected and has diameter at most 5. We also show how this result can be applied to derive information on the structure of a group from the normalizers of its Sylow subgroups.The second and third authors have been supported by Proyecto MTM2007-68010-C03-03 and Proyecto MTM2010-19938-C03-02, Ministerio de Educacion y Ciencia and FEDER, Spain.Kazarin, SL.; Martínez Pastor, A.; Pérez-Ramos, M. (2011). On the Sylow graph of a finite group. Israel Journal of Mathematics. 186(1):251-271. doi:10.1007/s11856-011-0138-xS2512711861Z. Arad and D. Chillag, Finite groups containing a nilpotent Hall subgroup of even order, Houston Journal of Mathematics 7 (1981), 23–32.H. Azad, Semi-simple elements of order 3 in finite Chevalley groups, Journal of Algebra 56 (1979), 481–498.A. Ballester-Bolinches, A. Martínez-Pastor, M. C. Pedraza-Aguilera and M. D. Pérez-Ramos, On nilpotent-like fitting formations, in Groups St. Andrews 2001 in Oxford, (C. M. Campbell et al., eds.) London Mathematical Society Lecture Note Series 304, Cambridge University Press, 2003, pp. 31–38.M. Bianchi, A. Gillio Berta Mauri and P. Hauck, On finite groups with nilpotent Sylow normalizers, Archiv der Mathematik 47 (1986), 193–197.A. Borel, R. Carter, C.W. Curtis, N. Iwahori, T. A. Springer, R. Steinberg, Seminar on Algebraic Groups and Related Finite Groups, Lecture Notes of Mathematics 131 Springer, Berlin, 1970.N. Bourbaki, Éléments de mathématique: Groupes et algèbres de Lie, Chapters IV, V, VI, Hermann, Paris, 1968.R. W. Carter, Simple groups of Lie type, Wiley, London, 1972.R. W. Carter, Conjugacy classes in the Weyl group, Compositio Mathematica 25 (1972), 1–59.R. W. Carter, Finite Groups of Lie Type: Conjugacy Classes and Complex Characters, Wiley, London, 1985.A. D’Aniello, C. De Vivo and G. Giordano, On certain saturated formations of finite groups, in Proceedings Ischia Group Theory 2006, (T. Hawkes, P. Longobardy and M. Maj, eds.) World Scientific, Hackensack, NJ, 2007, pp. 22–32.A. D’Aniello, C. De Vivo and G. Giordano, Lattice formations and Sylow normalizers: a conjecture, Atti del Seminario Matematico e Fisico dell’ Università di Modena e Reggio Emilia 55 (2007), 107–112.A. D’Aniello, C. De Vivo, G. Giordano and M. D. Pérez-Ramos, Saturated formations closed under Sylow normalizers, Communications in Algebra 33 (2005), 2801–2808.K. Doerk, T. Hawkes, Finite soluble groups, Walter De Gruyter, Berlin-New York, 1992.G. Glauberman, Prime-power factor groups of finite groups II, Mathematische Zeitschrift 117 (1970), 46–56.D. Gorenstein, R. Lyons, The local 2-structure of groups of characteristic 2 type, Memoirs of the American Mathematical Society 42, No. 276, Providence, RI, 1983.R. M. Guralnick, G. Malle and G. Navarro, Self-normalizing Sylow subgroups, Proceedings of the American Mathematical Society 132 (2004), 973–979.F. Menegazzo, M. C. Tamburini, A property of Sylow p-normalizers in simple groups, Quaderni del seminario Matematico di Brescia, n. 45/02 (2002).R. Steinberg, Lectures on Chevalley Groups, Yale University, New Haven, Conn., 1968.E. Stensholt, An application of Steinberg’s construction of twisted groups, Pacific Journal of Mathematics 55 (1974), 595–618.E. Stensholt, Certain embeddings among finite groups of Lie type, Journal of Algebra 53 (1978), 136–187.K. Zsigmondy, Zur Theorie der Potenzreste, Monatshefte für Mathematik and Physik 3 (1892), 265–284

Stereochemical and Skeletal Diversity Arising from Amino Propargylic Alcohols

An efficient synthetic pathway to the possible stereoisomers of skeletally diverse heterocyclic small molecules is presented. The change in shape brought about by different intramolecular cyclizations of diastereoisomeric amino propargylic alcohols is quantified using principal moment-of-inertia (PMI) shape analysis.Chemistry and Chemical Biolog

Where do you live? North versus Central-South differences in relation to Italian patients with oral lichen planus: a cross-sectional study from the SIPMO (Italian Society of Oral Pathology and Medicine)

Background Oral lichen planus (OLP) is an immune-mediated inflammatory chronic disease of the oral mucosa, with different patterns of clinical manifestations which range from keratotic manifestations (K-OLP) to predominantly non-keratotic lesions (nK-OLP). The aim of the study was to analyze the differences in the clinical, psychological profile and symptoms between Italian patients of the North and Central-South with K-OLP and nK-OLP. Methods 270 K-OLP and 270 nK-OLP patients were recruited in 15 Italian universities. The Numeric Rating Scale (NRS), Total Pain Rating Index (T-PRI), Hamilton Rating Scales for Depression and for Anxiety (HAM-D and HAM-A), Pittsburgh Sleep Quality Index (PSQI), and Epworth Sleepiness Scale (ESS) were administered. Results The Central-South K-OLP (CS-K-OLP) patients reported a higher frequency of pain/burning compared with the K-OLP patients of the North (N-K-OLP) with higher scores in the NRS and T-PRI (p value < 0.001**). The CS-K-OLP and the CS-nK-OLP patients showed higher scores in the HAM-D, HAM-A, PSQI and ESS compared with the Northern patients (p value < 0.001**). Multivariate logistic regression revealed that the NRS and T-PRI showed the greatest increase in the R2 value for the CS-K-OLP (DR2 = 9.6%; p value < 0.001**; DR2 = 9.7% p value < 0.001**; respectively) and that the oral symptoms (globus, itching and intraoral foreign body sensation) and PSQI showed the greatest increase in the R2 value for the CS-nK-OLP (DR2 = 5.6%; p value < 0.001**; DR2 = 4.5% p value < 0.001** respectively). Conclusions Pain and mood disorders are predominant in patients with OLP in the Central-South of Italy. Clinicians should consider that the geographical living area may explain the differences in oral symptoms and psychological profile in OLP

Targeted Inactivation of Cerberus Like-2 Leads to Left Ventricular Cardiac Hyperplasia and Systolic Dysfunction in the Mouse

Previous analysis of the Cerberus like 2 knockout (Cerl2(-/-)) mouse revealed a significant mortality during the first day after birth, mostly due to cardiac defects apparently associated with randomization of the left-right axis. We have however, identified Cerl2-associated cardiac defects, particularly a large increase in the left ventricular myocardial wall in neonates that cannot be explained by laterality abnormalities. Therefore, in order to access the endogenous role of Cerl2 in cardiogenesis, we analyzed the embryonic and neonatal hearts of Cerl2 null mutants that did not display a laterality phenotype. Neonatal mutants obtained from the compound mouse line Cer2(-/-)Fundacao para a Ciencia e Tecnologia (FCT); IBB/CBME [PEst-OE/EQB/LA0023/2011]; FCT [SFRH/BD/62081/2009]info:eu-repo/semantics/publishedVersio

Study of cosmogenic activation above ground for the DarkSide-20k experiment

The activation of materials due to exposure to cosmic rays may become an important background source for experiments investigating rare event phenomena. DarkSide-20k, currently under construction at the Laboratori Nazionali del Gran Sasso, is a direct detection experiment for galactic dark matter particles, using a two-phase liquid-argon Time Projection Chamber (TPC) filled with 49.7 tonnes (active mass) of Underground Argon (UAr) depleted in 39Ar. Despite the outstanding capability of discriminating / background in argon TPCs, this background must be considered because of induced dead time or accidental coincidences mimicking dark-matter signals and it is relevant for low-threshold electron-counting measurements. Here, the cosmogenic activity of relevant long-lived radioisotopes induced in the experiment has been estimated to set requirements and procedures during preparation of the experiment and to check that it is not dominant over primordial radioactivity; particular attention has been paid to the activation of the 120 t of UAr used in DarkSide-20k. Expected exposures above ground and production rates, either measured or calculated, have been considered in detail. From the simulated counting rates in the detector due to cosmogenic isotopes, it is concluded that activation in copper and stainless steel is not problematic. The activity of 39Ar induced during extraction, purification and transport on surface is evaluated to be 2.8% of the activity measured in UAr by DarkSide-50 experiment, which used the same underground source, and thus considered acceptable. Other isotopes in the UAr such as 37Ar and 3H are shown not to be relevant due to short half-life and assumed purification methods

Sensitivity projections for a dual-phase argon TPC optimized for light dark matter searches through the ionization channel

Dark matter lighter than 10  GeV/c2 encompasses a promising range of candidates. A conceptual design for a new detector, DarkSide-LowMass, is presented, based on the DarkSide-50 detector and progress toward DarkSide-20k, optimized for a low-threshold electron-counting measurement. Sensitivity to light dark matter is explored for various potential energy thresholds and background rates. These studies show that DarkSide-LowMass can achieve sensitivity to light dark matter down to the solar neutrino fog for GeV-scale masses and significant sensitivity down to 10  MeV/c2 considering the Migdal effect or interactions with electrons. Requirements for optimizing the detector’s sensitivity are explored, as are potential sensitivity gains from modeling and mitigating spurious electron backgrounds that may dominate the signal at the lowest energies

Study on cosmogenic activation above ground for the DarkSide-20k project

The activation of materials due to the exposure to cosmic rays may become an important background source for experiments investigating rare event phenomena. DarkSide-20k is a direct detection experiment for galactic dark matter particles, using a two-phase liquid argon time projection chamber filled with 49.7 tonnes (active mass) of Underground Argon (UAr) depleted in 39Ar. Here, the cosmogenic activity of relevant long-lived radioisotopes induced in the argon and other massive components of the set-up has been estimated; production of 120 t of radiopure UAr is foreseen. The expected exposure above ground and production rates, either measured or calculated, have been considered. From the simulated counting rates in the detector due to cosmogenic isotopes, it is concluded that activation in copper and stainless steel is not problematic. Activation of titanium, considered in early designs but not used in the final design, is discussed. The activity of 39Ar induced during extraction, purification and transport on surface, in baseline conditions, is evaluated to be 2.8% of the activity measured in UAr from the same source, and thus considered acceptable. Other products in the UAr such as 37Ar and 3H are shown to not be relevant due to short half-life and assumed purification methods