Finite groups with real conjugacy classes of prime size

We determine the structure of a finite group G whose noncentral real conjugacy classes have prime size. In particular, we show that G is solvable and that the set of the sizes of its real classes is one of the following: {1},{1, 2}, {1, p}, or {1, 2, p}, where p is an odd prime

Nonvanishing elements for Brauer characters

Let G be a finite group and p a prime. We say that a p-regular element g of G is p-nonvanishing if no irreducible p-Brauer character of G takes the value 0 on g. The main result of this paper shows that if G is solvable and g is a p-regular element which is p-nonvanishing, then g lies in a normal subgroup of G whose p-length and p'-length are both at most 2 (with possible exceptions for p\leq 7), the bound being best possible. This result is obtained through the analysis of one particular orbit condition in linear actions of solvable groups on finite vector spaces, and it generalizes (for p>7) some results in Dolfi and Pacifici [\u2018Zeros of Brauer characters and linear actions of finite groups\u2019, J. Algebra 340 (2011), 104\u2013113]

Assessment of real-time operative torque during nickel-titanium instrumentation with different lubricants

The aim of the present study is twofold: to assess ex vivo the role of different lubricants on real-time torque generated during intracanal instrumentation and to check whether two different kinds of torque parameters, operative torque (OT) and average peak torque (APT), could produce similar results. Forty extracted single-rooted teeth were selected for the present study and divided into four equal groups (n = 10): Group A, NaCl 0.2%; Group B, NaOCl 5%; Group C, ethylenediaminetetraacetate (EDTA), and Group D, EDTA and hydrogen peroxide. Afterwards, Edge Taper F2 (Edge Endo, Albuquerque, New Mexico) were rotated clockwise at 300 rpm with 3 Ncm maximum torque by an endodontic torque recording motor. In each sample, mean OT and mean APT were recorded and statistically analyzed with one-way ANOVA and a post hoc Bonferroni between groups (p < 0.05). EDTA (12.11 ± 4.45 Ncm) showed statistically significant (p < 0.05) lower values compared with the other tested irrigant for both parameters. Overall, the two different parameters were both able to differentiate between the influence of lubricants on torsional loads

On the vanishing prime graph of finite groups

Let G be a finite group. An element g 08 G is called a vanishing element of G if there exists an irreducible complex character \u3c7 of G such that \u3c7(g) = 0. In this paper we study the vanishing prime graph \u393(G), whose vertices are the prime numbers dividing the orders of some vanishing element of G, and two distinct vertices p and q are adjacent if and only if G has a vanishing element of order divisible by pq. Among other things we prove that, similarly to what holds for the prime graph of G, the graph \u393(G) has at most six connected components

On the orders of zeros of irreducible characters

Let G be a finite group and p a prime number. We say that an element g in G is a vanishing element of G if there exists an irreducible character χ of G such that χ(g)=0. The main result of this paper shows that, if G does not have any vanishing element of p-power order, then G has a normal Sylow p-subgroup. Also, we prove that this result is a generalization of some classical theorems in Character Theory of finite groups

On the vanishing prime graph of solvable groups

Let G be a finite group, and Irr(G) the set of irreducible complex characters of G. We say that an element g is an element of G is a vanishing element of G if there exists chi in Irr(G) such that chi(g) = 0. In this paper, we consider the set of orders of the vanishing elements of a group G, and we define the prime graph on it, which we denote by Gamma(G). Focusing on the class of solvable groups, we prove that Gamma(G) has at most two connected components, and we characterize the case when it is disconnected. Moreover, we show that the diameter of Gamma(G) is at most 4. Examples are given to round out our understanding of this matter. Among other things, we prove that the bound on the diameter is best possible, and we construct an infinite family of examples showing that there is no universal upper bound on the size of an independent set of Gamma(G)

On the character degree graph of finite groups

Given a finite group G, let cd (G) denote the set of degrees of the irreducible complex characters of G. The character degree graph of G is defined as the simple undirected graph whose vertices are the prime divisors of the numbers in cd (G) , two distinct vertices p and q being adjacent if and only if pq divides some number in cd (G). In this paper, we consider the complement of the character degree graph, and we characterize the finite groups for which this complement graph is not bipartite. This extends the analysis of Akhlaghi et al. (Proc Am Math Soc 146:1505\u20131513, 2018), where the solvable case was treated

Use of dermal-fat grafts in the post-oncological reconstructive surgery of atrophies in the zygomatic region: Clinical evaluations in the patients undergone to previous radiation therapy

Introduction: Grafting of autologous adipose tissue can be recommended in some cases of facial plastic surgery. Rhabdomyosarcoma is a type of cancer that can also affect the orbit. Enucleation of the eye can cause atrophy of the corresponding hemiface and decreased orbital growth.Case report: We report a case of a female patient with a medical history of surgical enucleation of the right eyeball, who had received rhabdomyosarcoma radiation therapy in her youth. The patient presented with a depression in the right zygomatic region. We took a dermal-fat flap from the abdominal region, which had been previously treated.Results: The surgical outcome, 48 hours, and much clearly 31 days after the surgery, revealed that the right zygomatic region had returned to its proper anatomical shape, although there were still signs of postoperative edema.Discussion: Very damaged tissues, like those exposed to radiation therapy, are generally not suitable for grafting of adipose tissue.Conclusions: In the described case, we achieved a technically and aesthetically satisfying result despite the patient's medical history involving several perplexities about the use of autologous dermal-fat tissues, because of prior radiation therapy exposure. The clinical case shows that even a region exposed to radiation therapy can be a valid receiving bed for dermal-fat grafting. © 2012 Inchingolo et al.; licensee BioMed Central Ltd

Finite groups with real-valued irreducible characters of prime degree

In this paper we describe the structure of finite groups whose real-valued nonlinear irreducible characters have all prime degree. The more general situation in which the real-valued irreducible characters of a finite group have all squarefree degree is also considered