相关论文: The $R_\infty$ property for nilpotent quotients of…

A finite group $G$ is said to have the nilpotent decomposition property (ND) if for every nilpotent element $\alpha$ of the integral group ring $\mathbb{Z}[G]$ one has that $\alpha e$ also belong to $\mathbb{Z}[G]$, for every primitive…

We will say that a group G possesses the Magnus property if for any two elements u,v in G with the same normal closure, u is conjugate to v or v^{-1}. We prove that some one-relator groups, including the fundamental groups of closed…

We study the groups $G$ with the curious property that there exists an element $k\in G$ and a function $f\colon G\to G$ such that $f(xk)=xf(x)$ holds for all $x\in G$. This property arose from the study of near-rings and input-output…

Let $G$ be a finite solvable group. We prove that if $\chi\in{\rm Irr}(G)$ has odd degree and $\chi(1)$ is the minimal degree of the non-linear irreducible characters of $G$, then $G/{\rm Ker} \chi$ is nilpotent-by-abelian.

Let $G$ be a simple simply-connected algebraic group over an algebraically closed field $k$ of characteristic $p>0$ with $\mathfrak{g}={\rm Lie}(G)$. We discuss various properties of nilpotent orbits in $\mathfrak{g}$, which have previously…

Let $S$ be a minimal surface of general type with irregularity $q(S) = 1$. Well-known inequalities between characteristic numbers imply that $3 p_g(S) \le c_2(S) \le 10 p_g(S)$, where $p_g(S)$ is the geometric genus and $c_2(S)$ the…

Let K be a field of positive characteristic p and KG the group algebra of a group G. It is known that if KG is Lie nilpotent then its upper (or lower) Lie nilpotency index is at most |G'|+1, where |G'| is the order of the commutator…

Let $p$ be a prime and let $G$ be a finite group such that the smallest prime that divides $|G|$ is $p$. We find sharp bounds, depending on $p$, for the commuting probability and the average character degree to guarantee that $G$ is…

For a simply connected solvable Lie group G with a cocompact discrete subgroup {\Gamma}, we consider the space of differential forms on the solvmanifold G/{\Gamma} with values in certain flat bundle so that this space has a structure of a…

Let $F$ be either a free nilpotent group of a given class and of finite rank or a free solvable group of a certain derived length and of finite rank. We show precisely which ones have the $R_{\infty}$ property. Finally, we also show that…

Let $R$ be a finite unitary ring such that $R=R_0[R^*]$ where $R_0$ is the prime ring and $R^*$ is not a nilpotent group. We show that if all proper subgroups of $R^*$ are nilpotent groups, then the cardinal of $R$ is a power of prime…

We associate a graph $\mathcal{N}_{G}$ with a group $G$ (called the non-nilpotent graph of $G$) as follows: take $G$ as the vertex set and two vertices are adjacent if they generate a non-nilpotent subgroup. In this paper we study the graph…

A group $G$ has property $R_\infty$ if for every $\phi\in Aut(G)$, there are an infinite number of $\phi$-twisted conjugacy classes of elements in $G$. In this note, we determine the $R_\infty$-property for $G=\pi_1(M)$ for all geometric…

Given a closed, oriented surface X of genus g>1, and a semisimple Lie group G, let R_G be the moduli space of reductive representations of the fundamental group of X in G. We determine the number of connected components of R_PGL(n,R), for…

A countable discrete group $\Gamma$ is said to have the relative ISR-property if for every non-trivial normal subgroup $N\trianglelefteq\Gamma$ and every von Neumann subalgebra $\mathcal{M}\subseteq L(\Gamma)$ invariant under conjugation by…

Let $F_g$ denote the closed orientable surface of genus $g$. What is the least order finite group, $G_g$, for which there is a homomorphism $\psi$ from $\pi_1(F_g)$ to $G_g$ so that no nontrivial simple closed curve on $F_g$ represents an…

The nilpotent graph of a group $G$ is the simple and undirected graph whose vertices are the elements of $G$ and two distinct vertices are adjacent if they generate a nilpotent subgroup of $G$. Here we discuss some topological properties of…

Let $R$ be a compact, connected, orientable surface of genus $g$, $Mod_R^*$ be the extended mapping class group of $R$, $\mathcal{C}(R)$ be the complex of curves on $R$, and $\mathcal{N}(R)$ be the complex of nonseparating curves on $R$. We…

This is the final paper in a series of four, concerning the surface $T \times T$ embedded in $\mathbb{CP}^8$, where $T$ is a the one dimensional torus. In this paper we compute the fundamental group of the Galois cover of the surface with…

Let G be a connected, compact, semisimple Lie group. It is known that for a compact closed orientable surface $\Sigma$ of genus $l >1$, the order of the group $H^2(\Sigma,\pi_1(G))$ is equal to the number of connected components of the…