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]

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

Aminosidine plus sodium stibogluconate for the treatment of Indian kala-azar: a randomized dose-finding clinical trial

This randomized, open sequential design trial was set up to assess the efficacy, tolerability and toxicity of 20 d courses of combined intramuscular aminosidine and sodium stibogluconate at various dosages in patients with newly-diagnosed kala-azar in Bihar, India. Three successive studies of 96 patients each were originally planned with aminosidine administered at 12, 6 and 3 mg/kg/d, respectively. For each aminosidine dosage, patients were randomly assigned to receive sodium stibogluconate at 20, 10 or 5 mg/kg/d of antimony. Ninety-six patients were enrolled and assigned aminosidine 12 mg/kg/d as scheduled. In the subsequent study with aminosidine at 6 mg/kg/d, the trial was interrupted after 40 patients had entered owing to inadequacy of the treatment. With aminosidine 12 mg/kg/d the success rates with sodium stibogluconate at 20, 10 and 5 mg/kg/d were 88%, 71% and 72%, respectively and did not differ significantly. With aminosidine 6 mg/kg/d, 69%, 50% and 46% of patients were cured with the same sodium stibogluconate doses, respectively; again, there was no significant difference between the subgroups. The overall success rate with aminosidine at 12 mg/kg/d (76%) was significantly higher than that with 6 mg/kg/d (55%) (odds ratio = 2·69; 95% confidence interval, 1·11-6·4). Patients improved clinically and the treatments were equally well tolerated. The combination of aminosidine 12 mg/kg/d and sodium stibogluconate 20 mg/kg/d for 20 d appears to be an effective and safe replacement in Bihar for sodium stibogluconate alone for ⩾40

Chiral spin liquid and emergent anyons in a Kagome lattice Mott insulator

Topological phases in frustrated quantum spin systems have fascinated researchers for decades. One of the earliest proposals for such a phase was the chiral spin liquid put forward by Kalmeyer and Laughlin in 1987 as the bosonic analogue of the fractional quantum Hall effect. Elusive for many years, recent times have finally seen a number of models that realize this phase. However, these models are somewhat artificial and unlikely to be found in realistic materials. Here, we take an important step towards the goal of finding a chiral spin liquid in nature by examining a physically motivated model for a Mott insulator on the Kagome lattice with broken time-reversal symmetry. We first provide a theoretical justification for the emergent chiral spin liquid phase in terms of a network model perspective. We then present an unambiguous numerical identification and characterization of the universal topological properties of the phase, including ground state degeneracy, edge physics, and anyonic bulk excitations, by using a variety of powerful numerical probes, including the entanglement spectrum and modular transformations.Comment: 9 pages, 9 figures; partially supersedes arXiv:1303.696

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 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 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