Related papers: Solving a group equation of length nine

Lehmer's totient problem asks whether there exists any composite number $n$ such that $\varphi(n) \, \mid \, (n-1)$, where $\varphi$ is Euler totient function. It is known that if any such $n$ exists, it must be Carmichael and $n >…

We study the class of 3-dimensional nonlinear 2-hessian equations mentioned in the text. We perform preliminary group classification on 2-hessian equation. In fact, we find additional equivalence transformation on the space (x,y,z,u,f),…

We study Brou\'e's abelian defect group conjecture for groups of Lie type using the recent theory of perverse equivalences and Deligne--Lusztig varieties. Our approach is to analyze the perverse equivalence induced by certain…

We consider Donovan's conjecture in the context of blocks of groups $G$ with defect group $D$ and normal subgroups $N \lhd G$ such that $G=C_D(D\cap N)N$, extending similar results for blocks with abelian defect groups. As an application we…

Recently, there has been substantial progress on the Alperin weight conjecture. As a step to establish the Alperin weight conjecture for all finite groups, we prove the inductive blockwise Alperin weight condition for simple groups of…

We discuss many surprising implications of a positive answer to a question raised in some cases by Greenberg in the $`70$s and more generally by Shalom in the early $2000$s. We refer to this positive answer as the Greenberg-Shalom…

Yangming Li and Xianhua Li in 2012 proposed a conjecture that generalizes O.U. Kramer's result about supersoluble groups. Here we proved that this conjecture is false in the general case and true for groups with the trivial Frattini…

We introduce a notion of metric on a Lie groupoid, compatible with multiplication, and we study its properties. We show that many families of Lie groupoids admit such metrics, including the important class of proper Lie groupoids. The…

A 1930s conjecture of Hopf states that an even-dimensional compact Riemannian manifold with positive sectional curvature has positive Euler characteristic. We prove this conjecture under the additional assumption that the isometry group has…

In 1977 Pohst conjectured a certain inequality for $n$ variables and give a computer-assisted proof for $n\leq 10$. We give a proof for all $n$ using a combinatorial argument. This inequality yields a better bound for the regulator in terms…

Given a group $G$ with bounded torsion that acts properly on a systolic complex, we show that every solvable subgroup of $G$ is finitely generated and virtually abelian of rank at most $2$. In particular this gives a new proof of the above…

In 1934 N. N. Luzin proved in his short (but dense) paper \textit{Sur la decomposition des ensembles} that every set $X\subseteq \mathbb{R}$ can be decomposed into two full, with respect to Lebesgue measure or category, subsets. We will try…

A group is coherent if all its finitely generated subgroups are finitely presented. In this article we provide a criterion for positively determining the coherence of a group. This criterion is based upon the notion of the perimeter of a…

Artin conjectured that certain Galois representations should give rise to entire L-series. We give some history on the conjecture and motivation of why it should be true by discussing the one-dimensional case. The first known example to…

Using the classification of finite simple groups we prove Alperin's weight conjecture and the character theoretic version of Broue's abelian defect group conjecture for 2-blocks of finite groups with an elementary abelian defect group of…

We use the Bateman--Horn Conjecture from number theory to give strong evidence of a positive answer to Peter Neumann's question, whether there are infinitely many simple groups of order a product of six primes. (Those with fewer than six…

The Levin method is a well-known technique for evaluating oscillatory integrals, which operates by solving a certain ordinary differential equation in order to construct an antiderivative of the integrand. It was long believed that this…

Fundamental conjectures in modular representation theory of finite groups, more precisely, Alperin's Weight Conjecture and Robinson's Ordinary Weight Conjecture, can be expressed in terms of fusion systems. We use fusion systems to connect…

The notion of acyclic matching property was provided by Losonczy and it was proved that torsion-free groups admit this property. In this paper, we introduce a duality of acyclic matching as a tool for classification of some Abelian groups,…

The study of Fuglede's conjecture on the direct product of elementary abelian groups was initiated by Iosevich et al. For the product of two elementary abelian groups the conjecture holds. For $\mathbb{Z}_p^3$ the problem is still open if…