Related papers: Principal blocks for different primes, II

We prove that finite-dimensional Jacobian algebras associated with non-degenerate quivers with potentials satisfy the stable Brauer-Thrall II' conjecture. In particular, this implies that the brick Brauer-Thrall II' conjecture (also known…

We give a new approach to the construction of derived equivalences between blocks of finite groups, based on perverse equivalences, in the setting of Brou\'e's conjecture. We provide in particular local and global perversity data describing…

Problem 21 of Brauer's list of problems from 1963 asks whether for any positive integer k there are finitely many isomorphism classes of groups that occur as the defect group of a block with k irreducible characters. We solve this problem…

We have found a list of finite simple groups with cyclic Sylow $p$-subgroup whose principal $p$-blocks have Brauer trees in the shape of a star, that is a tree of diameter at most $2$. Moreover, for an arbitrary finite group $G$ with cyclic…

Brauer and Fowler noted restrictions on the structure of a finite group G in terms of the order of the centralizer of an involution t in G. We consider variants of these themes. We first note that for an arbitrary finite group G of even…

In the paper, there are new found methods to determine the range of every exceptional element in exceptional set, we can solve Twin primes problem and Goldbach Conjecture problem basically.

Trinh and Xue have proposed a startling conjecture on intersections of blocks of cyclotomic Hecke algebras occurring in modular representation theory of finite reductive groups. We prove this conjecture for all exceptional type groups apart…

By generalizing Frobenius' polynomial method to good partition algebra, we will develop new character theories for a finite group $G$. A uniform defining equations are derived for these kinds of character theories. The new character…

A finite group $G$ is coprimely-invariably generated if there exists a set of generators $\{g_1, \ldots, g_d\}$ of $G$ with the property that the orders $|g_1|, \ldots, |g_d|$ are pairwise coprime and that for all $x_1, \ldots, x_d \in G$…

Let $G$ be a flat finite-type group scheme over a scheme $S$, and $X$ a noetherian $S$-scheme on which $G$-acts. We define and study $G$-prime and $G$-primary $G$-ideals on $X$ and study their basic properties. In particular, we prove the…

Let $G$ be a finite group acting vertex-transitively on a graph. We show that bounding the order of a vertex stabilizer is equivalent to bounding the second singular value of a particular bipartite graph. This yields an alternative…

We generalise the Third Main Theorem by Brauer, the First and Second Fong Reduction to generalised block fusion systems and apply the Second Fong Reduction to extend a result by Cabanes about the non-exoticity of fusion systems of unipotent…

We establish a conjectured result of Dummit, Granville and Kisilevsky for the maximum bias in P2 races, which compare the number of P2's $\leq x$ whose prime factors, evaluated at a given quadratic character, take specifc values.

This article is a survey based on our earlier paper ("The 'Vertical' Generalization of the Binary Goldbach's Conjecture as Applied on 'Iterative' Primes with (Recursive) Prime Indexes (i-primeths)" [11]), a paper in which we have proposed a…

We propose a projective version of the celebrated Brauer's Height Zero Conjecture on characters of finite groups and prove it, among other cases, for $p$-solvable groups as well as for (some) quasi-simple groups.

In 1987, the second author of this paper reported his conjecture, all finite simple groups $S$ can be characterized uniformly using the order of $S$ and the set of element orders in $S$, to Prof. J. G. Thompson. In their communications,…

Let $G$ be a group with identity element $e$, and suppose that $S$ is an associative $G$-graded ring that is not necessarily unital. In the case where $G$ is an ordered group, we show that a graded ideal is prime if and only if it is graded…

This paper introduces a new method to find the next prime number after a given prime ${P}$. The proposed method is used to derive a system of inequalities, that serve as constraints which should be satisfied by all primes whose successor is…

We study the operator-valued positive definite functions on a group using positive block matrices. We give an alternative proof to Brehmer positivity for doubly commuting contractions. We classify all commuting unitary representations over…

In this article we calculate two aspects of the representation theory of a Brauer configuration algebra: its Cartan matrix, and the module length of its associated indecomposable projective modules. Then we introduce the concept of…