# Irreducible complex character degree

## Top PDF Irreducible complex character degree: ### A new characterization of Mathieu groups by the order and one irreducible character degree

Classifying ﬁnite groups by the properties of their characters is an interesting problem in group theory. In , Huppert conjectured that each ﬁnite non-abelian simple group G is characterized by the set cd(G) of degrees of its complex irreducible characters. In [–], it was shown that many non-abelian simple groups such as L  (q) and S z (q) satisfy ### On nonsolvable groups whose prime degree graphs have four vertices and one triangle

Let G be a finite group. We consider the set of the irreducible complex characters of G, namely Irr(G), and the related degree set cd(G) = { χ(1) : χ ∈ Irr(G) } . Let ρ(G) be the set of all primes which divide some character degree of G. There is a large literature which is devoted to study the ways in which one can associate a graph with a group, for the purpose of investigating the algebraic structure using properties of the associated graph. One of these graphs is the prime degree graph of G which is denoted by ∆(G). It is an undirected graph with vertex set ρ(G), where p, q ∈ ρ(G) are joined by an edge if there exists an irreducible character degree χ(1) ∈ cd(G) \ { 1 } which is divisible by pq. ### Characterization of some simple \$K_4\$-groups by some irreducible complex character degrees

Throughout this paper, let G be a finite group and let all characters be complex characters. Also, let l(G) be the largest irreducible character degree of G, s(G) be the second largest irreducible character degree of G and t(G) be the third largest irreducible character degree of G. The set of all irreducible characters of G is shown by Irr(G) and the set of all irreducible character degrees of G is shown by cd(G). In , B. Huppert conjectured that if G is a finite group and S is a finite non-abelian simple group such that cd(G) = cd(S), then G ∼ = S × A, where A is an abelian group. In ,  and , it is shown that L 2 (p), simple K 3 -groups and Mathieu simple groups are determined uniquely by their ### Bipartite divisor graph for the set of irreducible character degrees

Abstract. Let G be a finite group. We consider the set of the irreducible complex characters of G, namely Irr(G), and the related degree set cd(G) = {χ(1) : χ ∈ Irr(G)}. Let ρ(G) be the set of all primes which divide some character degree of G. In this paper we introduce the bipartite divisor graph for cd(G) as an undirected bipartite graph with vertex set ρ(G) ∪ (cd(G) \ {1}), such that an element p of ρ(G) is adjacent to an element m of cd(G) \ { 1 } if and only if p divides m. We denote this graph simply by B(G). Then by means of combinatorial properties of this graph, we discuss the structure of the group G. In particular, we consider the cases where B (G) is a path or a cycle. ### Classifying families of character degree graphs of solvable groups

of vertices, of size k and t, respectively), and we place an edge between the two graphs injectively. That is, we attach edges uniquely from one complete graph to the other in a one-to-one fashion. The construction clearly gives a graph that satisfies P´ alfy’s condition and is at most diameter two. However, when determining if the graph can or cannot occur as the prime character degree graph of a solvable group, many of our arguments rely on facts about graphs that are diameter three. This has been studied more extensively in  and , and their techniques and results will play an important role. ### Recent progress on the elliptic curve discrete logarithm problem

An approach that looks promising is to study the so-called first-fall degree (FFD) of the polynomial systems. A Gr¨ obner-basis algorithm like F4 or F5 proceeds with polynomials of increasing degree while computing the basis, maybe backtracking to handle smaller degree polynomials if they happen to be created along the process. Roughly speaking, the FFD is the degree currently handled by the algorithm when this backtrack occurs for the first time. It has been observed that this degree is often not too far from the maximal degree that will ever occur in the algorithm. This is the first fall degee assumption (FFDA). Therefore, being able to bound the FFD, and assuming that the FFDA holds for the systems involved in ECDLP, we could dream to get better estimates for the whole computation. ### Actions of groups of birationally extendible automorphisms

For a non-compact subgroup G ⊂ Aut(D) the situation is substantially different. First of all, as is seen in the simplest example of the disk D in the complex plane, it is rational functions which play an important role. Secondly, since orbits are non-compact, one is led to study the action near the boundary. Without loss of generality we may assume that G is closed in Aut(D) so that it acts properly and let p ∈ ∂D be in the closure of some orbit z ∈ D. The geometry of the action near p is extremely rich. In fact, under reasonable regularity assumptions, it might happen that knowledge of the local action near p determines D itself and general classification results can be proved. There are numerous indications of this (see e.g. [8,17,21,22,31,32]) with Rosay’s Theorem being the easiest to state: If p is a strongly pseudoconvex boundary point, then D is biholomorphically equivalent to the unit ball B n := { P |z i | 2 < 1}. Under ### Finding the Galois Group of a polynomial: a demonstration of Stauduhar’s Method

A BSTRACT . The purpose of this paper is to demonstrate an algorithm to find the Galois group of any monic irreducible polynomial over the field of the rationals with integer coefficients. This algorithm was invented by Richard Stauduhar , hence, for the rest it is called the Stauduhar’s method. In order to identify the correct subgroup of S n , several conditions are assumed. Since in every case complex roots, discriminants, ### CT and Angiography of Primary Extradural Juxtasellar Tumors

The tumors in our series could be differentiated by the presence or absence of tumor calcification , degree and character of contrast enhancement, degree and location of bone erosion, an[r] ### Orthant spanning simplexes with minimal volume

Speciﬁcally, in this paper, it has been proved that the solution of the opti- mal simplex problem depends on the positive root of a (2 n − 1)-degree poly- nomial. This polynomial cannot be solved using radicals for any n from 3 up to 15 when the coordinates of A are transcendental over Q . It is likely that it cannot be solved by radicals for any n, although a proof has not been found. Limited to dimension n = 3 and points A of distinct integer coordinates, it has been shown that if the polynomial p 3 (t) of degree 7 is irreducible, then it is ### A scheme over quasi-prime spectrum of modules

2. A. Azizi, Strongly irreducible ideals, J. Aust. Math. Soc. 84 (2008), 145-154. 3. Bourbaki, N. Commutative Algebra, Chap. 1-7, Paris: Hermann, (1972). 4. R. Hartshorne, Algebraic Geometry, Springer-Verlag New York Inc, (1977). 5. W. J. Heinzer and L. J. Ratlif and D. E. Rush, Strongly irreducible ideals of a ### A Case Study of Modeling Technique on Building a Smart Chinese International Learning Environment

completed the self-adapting education system concerning complex sentence types. It will continue to construct all other self-adapting systems for International Chinese language learning, hoping to achieve effective construction of a complete learning environment for Smart Chinese Learning. With the help of the Learning analytics technology, teachers have more opportunity to discover the real learning situation of each student. The Smart Learning environment can help students to assess their academic progress, predict their future performance, discover their potential problems and get suggestions. Therefore, Smart Learning can provide high-quality, personalized educational services to every student and raise intelligent, well educated people. ### Chronic irreducible dislocation of the proximal interphalangeal joint of the fifth toe: a case report

Case presentation: We report the case of a 47-year-old woman who sustained a minor foot injury for more than 30 years, resulting in chronic, irreducible dislocation of the proximal interphalangeal joint of the fifth toe. The affected proximal interphalangeal joint was accessed via a dorsal incision over the unstable interphalangeal joint. It was found that the interposed interphalangeal joint capsule and attenuated lateral collateral ligament were reconstructed, and it was stabilized by temporary insertion of a Kirschner wire. The affected joint was found to be stable, well-positioned and pain-free at the 12-month post-surgical check-up. ### Computing in Algebraic Closures of Finite Fields

The system described in  uses linear algebra to describe the embeddings of finite fields. From a complexity point of view, this is far from optimal: one may hope to compute and apply the morphisms in quasi-linear time in the degree of the extension, but this is usually out of reach of linear algebra techniques. Even worse, the quadratic memory requirements make the system unsuitable for embeddings of large degree extensions. Although the Magma core has evolved since the publication of the paper, experiments in Section 4.5 show that embeddings of large extension fields are still out of reach. ### ALEXANDER I N V A R I A N T S O F H Y P E R S U R FA C E COM P L E M E N T S

In their attempt of understanding the relation between the local and global topo- logical structure of stratified spaces, Cappell and Shaneson () investigated the invariants associated with a stratified pseudomanifold X, PL-embedded in codimen- sion two in a manifold Y (e.g., X might be a hypersurface in a smooth algebraic variety). In describing the L-classes of the subspace X, they use as a main tool the peripheral complex R • , a torsion, self-dual, perverse sheaf, supported on X (here ### Complex Word Identification Using Character n grams

2017), the second shared task has been organ- ised at the BEA workshop 2018 (Yimam et al., 2018) featuring a multilingual dataset. The dataset consists of training and testing sets for three lan- guages: English, German and Spanish, as well as French test set for cross-lingual CWI. The goal was to predict which words could be difficult for a non-native speaker, based on annotations collected from a mixture of native and non-native speakers. The predictions could be submitted in the form of class labels (complex or simple) and/or in the form of complexity probabilities. ### Primitive irreducible linear groups

So {Qijl is a system of imprimitivity for G, as claimed. Since elements of G permute the cosets of H^, suitably ordered, in the same way as they permute the subspaces Q^^, G is embedded in the permutational wreath product of H-j_ by 00, where 0 is as in the first line of this proof, via the Frobenius embedding. Now H^ has a representation on p(n/k) (namely h^^ h ^ ); it follows that H^ Wr GO has a representation on in which each copy of H^ acts on a copy of and the gO*s permute the copies of in the same way that they permute the copies of H^ in the wreath product. According to Kovacs 196?, this representation is (isomorphic to) that induced from the representation of H^ referred to above. But, by Curtis and Reiner 1962, (50.2), the representation induced from that of H^ is equivalent to the original faithful irreducible imprimitive representation of G on ### Improving the Polynomial time Precomputation of Frobenius Representation Discrete Logarithm Algorithms - Simplified Setting for Small Characteristic Finite Fields

given in (4). Assuming that both g 1 and g 2 are not equal to v + w then all three values (g 1 , g 2 , g 3 ) are well-defined. With this specific choice, the right-hand side that now appear in Equation (5) or (6) gains a new systematic degree 2 factor θh 1 + h 0 + (v + w)h 1 = (1 + u) θ 2 + (1 + u)(v + w)θ + vw + v 2 as given in Lemma 4. Again, the remaining factor in the right-hand side when considering these groups is of degree 4. Since the probability of a degree 4 polynomial to factor in terms of degree at most 2 is higher than 1/3, we can recover all the discrete logarithms of the irreducible polynomials of P g 1 ,G(g 1 ),G(G(g 1 )) .  