Related papers: Higher Congruences in Character Tables

Let $G$ be a finite Chevalley group. We are concerned with computing the values of the unipotent characters of $G$ by making use of Lusztig's theory of character sheaves. In this framework, one has to find the transformation between several…

Let $N(G)$ be the set of conjugacy classes sizes of $G$. We prove that if $N(G)=\Omega\times \{1,n\}$ for specific set $\Omega$ of integers, then $G\simeq A\times B$ where $N(A)=\Omega$, $N(B)=\{1,n\}$, and $n$ is a power of prime.

Let G be a semisimple affine algebraic group over a field F. Assuming that G becomes of inner type over some finite field extension of F of degree a power of a prime p, we investigate the structure of the Chow motives with coefficients in a…

To determine whether a number is congruent or not is an old and difficult topic and progress is slow. The paper presents a new theorem when a prime number is a congruent number or not. The proof is not necessarily any simpler or shorter…

Let $G$ be a finite group and let $\pi$ be a set of primes. Write $\mathrm{Irr}_{\pi'}(G)$ for the set of irreducible characters of degree not divisible by any prime in $\pi$. We show that if $\pi$ contains at most two prime numbers and the…

Let $R$ be a finite unital commutative ring. We introduce a new class of finite groups, which we call hereditary groups over $R$. Our main result states that if $G$ is a hereditary group over $R$ then a unital algebra isomorphism between…

For a finite group $G$, the vertices of the prime graph $\Gamma(G)$ are the primes that divide $|G|$, and two vertices $p$ and $q$ are connected by an edge if there is an element of order $pq$ in $G$. Prime graphs of solvable groups have…

We study the permutation action of a finite symplectic group of characteristic 2 on the set of subspaces of its standard module which are either totally isotropic or else complementary to totally isotropic subspaces with respect to the…

Uniform interpolation properties are defined for equational consequence in a variety of algebras and related to properties of compact congruences on first the free and then the finitely presented algebras of the variety. It is also shown,…

Let v(n) denote the number of compositions (ordered partitions) of a positive integer n into powers of 2. It appears that the function v(n) satisfies many congruences modulo 2^N. For example, for every integer B there exists (as k tends to…

We derive some equalities for relations on the algebra A, under the assumption that every subalgebra of A $\times$ A is congruence modular.

It is well known that if two finite groups have the same symmetric tensor categories of representations over C, then they are isomorphic. We study the following question: when do two finite groups G1,G2 have the same tensor categories of…

We investigate a beautiful conjecture of T. Wilde on character values and element orders of finite groups. We reduce it to a statement on nearly simple groups that can be checked ``prime by prime". For these groups, we show that a strong…

We investigate the problem asking when any square matrix whose entries lie in a finite field of characteristic 2 is decomposable into the sum of a diagonalizable matrix and a nilpotent matrix with index of nilpotency at most 2 and, as a…

We present a method for obtaining congruences modulo powers of a prime number~$p$ for combinatorial sequences whose generating function satisfies an algebraic differential equation. This method generalises the one by Kauers and the authors…

There has been some interest on how the average character degree affects the structure of a finite group. We define, and denote by $ \mathrm{anz}(G) $, the average number of zeros of characters of a finite group $ G $ as the number of zeros…

Let k be a field of characteristic p>0, and G be a finite group. The first result of this paper is an explicit formula for the determinant of the Cartan matrix of the Mackey algebra mu_k(G) of G over k. The second one is a formula for the…

We investigate the finite groups $G$ for which $\chi(1)^{2}=|G:Z(\chi)|$ for all characters $\chi \in Irr(G)$ and $|cd(G)|=2$, where $cd(G)=\{\chi(1)| \chi \in Irr(G)\}$. We call such a group a GVZ-group with two character degrees. We…

The prime graph of a finite group $G$ is the labelled graph $\Gamma(G)$ with vertices the prime divisors of $|G|$ and edges the pairs $\{p,q\}$ for which $G$ contains an element of order $pq$. A group $G$ is recognisable by its prime graph…

In this paper we show that the irreducible representations of a finite inverse semigroup $S$ over an algebraically closed field $F$ are in bijection with the conjugacy classes of $S$ if the characteristic of $F$ is zero or a prime number…