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Let M be a filtered module. Some properties of elements of M are "generic" in the following sense: (being open/stable) if an element z of M has a property P then any approximation of z has P; (being dense) any element of M is approximated…

Let $G$ be a finite group and $N_{\Omega}(G)$ be the intersection of the normalizers of all subgroups belonging to the set $\Omega(G),$ where $\Omega(G)$ is a set of all subgroups of $G$ which have some theoretical group property. In this…

In this report we summarize this work, all finite simple groups $G$ can determined uniformly using their orders $|G|$ and the set $\pi_e(G)$ of their element orders.

Let $C(T)$ be a generalized Coxeter group, which has a natural map onto one of the classical Coxeter groups, either $B_n$ or $D_n$. Let $C_Y(T)$ be a natural quotient of $C(T)$, and if $C(T)$ is simply-laced (which means all the relations…

As defined by Guralnick and Saxl given a nonabelian simple group $S$ and its nonidentity automorphism $x$, a natural number $\alpha_S(x)$ does not exceed a natural number $m$ if some $m$ conjugates of $x$ in the group $\langle x,S\rangle$…

In the paper we consider the following conjecture: if a finite group $G$ possesses a solvable $\pi$-Hall subgroup $H$, then there exist elements $x,y,z,t\in G$ such that the identity $H\cap H^x\cap H^y\cap H^z\cap H^t=O_\pi(G)$ holds. The…

We prove the Tits-Weiss conjecture for Albert division algebras over fields of arbitrary characteristics in the affirmative. The conjecture predicts that every norm similarity of an Albert division algebra is a product of a scalar homothety…

Groups definable in simple theories retain the chain conditions and decomposition properties known from stable groups, up to commensurability. In the small case, if a generic type of G is not foreign to some type q, there is a q-internal…

We give an explicit description of the set of element orders for every almost simple group with socle $E_7(q)$.

A celebrated theorem of Selberg states that for congruence subgroups of SL(2,Z) there are no exceptional eigenvalues below 3/16. We prove a generalization of Selberg's theorem for infinite index "congruence" subgroups of SL(2,Z).…

Motivated by Lusztig's $G$-stable pieces, we consider the combinatorial pieces: the pairs $(w, K)$ for elements $w$ in the Weyl group and subsets $K$ of simple reflections that are normalized by $w$. We generalize the notion of cyclic shift…

We give a new proof of Quillen's conjecture for solvable groups via a geometric and explicit method. For p-solvable groups, we provide both a new proof using the Classification of Finite Simple Groups and an asymptotic version without…

Let $G$ be a connected linear algebraic group over a number field $K$, let $\Gamma$ be a finitely generated Zariski dense subgroup of $G(K)$ and let $Z\subseteq G(K)$ be a thin set, in the sense of Serre. We prove that, if…

The Oort conjecture (now a theorem of Obus-Wewers and Pop) states that if k is an algebraically closed field of characteristic p, then any cyclic branched cover of smooth projective k-curves lifts to characteristic zero. This is equivalent…

If G is a complex semisimple algebraic group, we characterize the normality and the smoothness of its simple linear compactifications, namely those equivariant GxG-compactifications which possess a unique closed orbit and which arise in a…

Let $G$ be a finite group, and let $\kappa(G)$ be the probability that elements $g$, $h\in G$ are conjugate, when $g$ and $h$ are chosen independently and uniformly at random. The paper classifies those groups $G$ such that $\kappa(G) \geq…

This paper introduces the concept of slender generalized groups, extending the classical notion of slender abelian groups to the setting of generalized groups (completely simple semigroups). We establish fundamental properties of slender…

The Gruenberg-Kegel graph $\Gamma(G)$ of a finite group $G$ is the graph whose vertex set is the set of prime divisors of $|G|$ and in which two distinct vertices $r$ and $s$ are adjacent if and only if there exists an element of order $rs$…

Given a $\Gamma$-semigroup $S$, we construct a semigroup $\Sigma$ in such a way that one sided ideals and quasi-ideals of $S$ can be regarded as one sided ideals and quasi-ideals respectively of $\Sigma$. This correspondence and other…

In this (partly expository) paper we show, using ideas from the theory of total positivity, how a number of properties of a semisimple group over the complex numbers can be presented purely in terms of the Weyl group. We also describe some…