Related papers: The reduction theorem for relatively maximal subgr…

The group $\mathfrak{X}(G)$ is obtained from $G\ast G$ by forcing each element $g$ in the first free factor to commute with the copy of $g$ in the second free factor. We make significant additions to the list of properties that the functor…

Let G be a reductive linear algebraic group over an algebraically closed field of characteristic p > 0. A subgroup of G is said to be separable in G if its global and infinitesimal centralizers have the same dimension. We study the…

Given a group $G$, we write $g^G$ for the conjugacy class of $G$ containing the element $g$. A theorem of B. H. Neumann states that if $G$ is a group in which all conjugacy classes are finite with bounded size, then the commutator subgroup…

Landau's theorem on conjugacy classes asserts that there are only finitely many finite groups, up to isomorphism, with exactly $k$ conjugacy classes for any positive integer $k$. We show that, for any positive integers $n$ and $s$, there…

A theorem of L\"utkebohmert states that a rigid group homomorphism from the formal multiplicative group to a smooth commutative rigid group $G$, with relatively compact image, can be extended to a homomorphism from the rigid multiplicative…

A relative one-relator presentation has the form P = < X,H ; R > where X is a set, H is a group, and R is a group word on X and H. We show that if the group word on X obtained from R by deleting all the terms from H has what we call the…

Let $X$ be a nonempty real variety that is invariant under the action of a reflection group $G$. We conjecture that if $X$ is defined in terms of the first $k$ basic invariants of $G$ (ordered by degree), then $X$ meets a $k$-dimensional…

We prove the existence of a regular semigroup F(X) weakly generated by X such that all other regular semigroups weakly generated by X are homomorphic images of F(X). The semigroup F(X) is introduced by a presentation and the word problem…

A group $G$ is called subgroup conjugacy separable if for every pair of non-conjugate finitely generated subgroups of $G$, there exists a finite quotient of $G$ where the images of these subgroups are not conjugate. We prove that limit…

Let $A$ be an excellent two-dimensional normal local ring containing an algebraically closed field and let $X\to \mathrm{Spec} (A)$ be a resolution of singularity. We prove a theorem giving a condition under which the dimension of the…

We show that the finiteness length of an $S$-arithmetic subgroup $\Gamma$ in a noncommutative isotropic absolutely almost simple group $G$ over a global function field is one less than the sum of the local ranks of $G$ taken over the places…

We consider the group $\mathfrak{X}(G)$ obtained from $G\ast G$ by forcing each element $g$ in the first free factor to commute with the copy of $g$ in the second free factor. Deceptively complicated finitely presented groups arise from…

Let $\sigma=\{\sigma_j\,:\, j\in J\}$ be a partition of the set $\mathbb{P}$ of all prime numbers. A subgroup $X$ of a finite group $G$ is~\textit{$\sigma$-subnormal} in $G$ if there exists a chain of subgroups $$X=X_0\leq X_1\leq\ldots\leq…

Given a group $G$, we write $x^G$ for the conjugacy class of $G$ containing the element $x$. A famous theorem of B. H. Neumann states that if $G$ is a group in which all conjugacy classes are finite with bounded size, then the derived group…

A subgroup $H$ of a group $G$ is confined if the $G$-orbit of $H$ under conjugation is bounded away from the trivial subgroup in the space $\operatorname{Sub}(G)$ of subgroups of $G$. We prove a commutator lemma for confined subgroups. For…

Let for a prime $p$, $\mathfrak{X}$ (respectively $\mathfrak{Y}$) be the class of all $p$-biprimitively finite (respectively periodic $p$-conjugatively biprimitively finite) groups and $G\in \mathfrak{X}$ (respectively $G\in \mathfrak{Y}$),…

Let G be a finite abelian group of torsion r and let A be a subset of G. The Freiman--Ruzsa theorem asserts that if |A+A| < K|A| then A is contained in a coset of a subgroup of G of size at most r^{K^4}K^2|A|. It was conjectured by Ruzsa…

Let $G$ be a finite non-solvable group. We prove that there exists a proper subgroup $A$ of $G$ such that $G$ is the product of three conjugates of $A$, thus replacing an earlier upper bound of $36$ with the smallest possible value. The…

We prove Schlichting's theorem for approximate subgroups: if $\mathcal{X}$ is a uniform family of commensurable approximate subgroups in some ambient group, then there exists an invariant approximate subgroup commensurable with…

The aim of this paper is to give a proof of the restriction theorems for principal bundles with a reductive algebraic group as structure group in arbitrary characteristic. Let $G$ be a reductive algebraic group over any field $k=\bar{k}$,…