Related papers: On $T$-sequences and characterized subgroups

Let $\mathfrak F$ be a formation and let $G$ be a group. A subgroup $H$ of $G$ is $\mathrm{K}\mathfrak F$-subnormal (submodular) in $G$ if there is a subgroup chain $H=H_0\le \ H_1 \le \ \ldots \le H_i \leq H_{i+1}\le \ldots \le \ H_n=G$…

Let $G$ be a locally compact topological group, $G_0$ the connected component of its identity element, and comp(G) the union of all compact subgroups. A topological group will be called inductively monothetic if any subgroup generated (as a…

Let $A$ be an elementary abelian $r$-group with rank at least $3$ that acts faithfully on the finite $r'$-group $G$. Assume that $G$ is $A$-simple, so that $G = K_{1} \times\cdots\times K_{n}$ where $K_{1},\ldots,K_{n}$ is a collection of…

We prove two results about the natural representation of a group G of automorphisms of a normal projective threefold X on its second cohomology. We show that if X is minimal then G, modulo a normal subgroup of null entropy, is embedded as a…

A sequence of integers $ \{ s_n \}_{n \in \mathbb{N}} $ is called a T-sequence if there exists a Hausdorff group topology on $ \mathbb{Z} $ such that $ \{ s_n \}_{n \in \mathbb{N}} $ converges to zero. For every finite set of primes $ S $…

The focus of this paper is to better understand the coexistence of rigidity, weak mixing, and recurrence by constructing thin sets in the product of countably many copies of the finite cyclic group of order q. A Kronecker-type set K is a…

In this paper we investigate hereditarily normal topological groups and their subspaces. We prove that every compact subspace of a hereditarily normal topological group is metrizable. To prove this statement we first show that a…

A topological gyrogroup is a gyrogroup endowed with a topology such that the binary operation is jointly continuous and the inverse mapping is also continuous. It is shown that each compact subset of a topological gyrogroup with an…

Let $G$ be a simple, simply connected algebraic group over the field of complex numbers. We give a necessary and a sufficient condition for a Schubert variety $X(\tau)$ for which all the higher cohomologies $H^{i}(X(\tau), E)$ vanish for…

For an automorphism group G on an n-dimensional (n > 2) normal projective variety or a compact K\"ahler manifold X so that G modulo its subgroup N(G) of null entropy elements is an abelian group of maximal rank n-1, we show that N(G) is…

We generalize two of our previous results on abelian definable groups in $p$-adically closed fields to the non-abelian case. First, we show that if $G$ is a definable group that is not definably compact, then $G$ has a one-dimensional…

Greenberg proved that every countable group $A$ is isomorphic to the automorphism group of a Riemann surface, which can be taken to be compact if $A$ is finite. We give a short and explicit algebraic proof of this for finitely generated…

Let $G$ be a locally compact totally disconnected topological group. Under a necessary mild assumption, we show that the irreducible unitary representations of $G$ are uniformly admissible if and only if the irreducible smooth…

We prove that, if $G$ is a second-countable topological group with a compatible right-invariant metric $d$ and $(\mu_{n})_{n \in \mathbb{N}}$ is a sequence of compactly supported Borel probability measures on $G$ converging to invariance…

Let x be an element of a group G. For a positive integer n let E_n(x) be the subgroup generated by all commutators [...[[y,x],x],...,x] over y in G, where x is repeated n times. There are several recent results showing that certain…

For a finite group $G$, we define the inclusion graph of subgroups of $G$, denoted by $\mathcal I(G)$, is a graph having all the proper subgroups of $G$ as its vertices and two distinct vertices $H$ and $K$ in $\mathcal I(G)$ are adjacent…

We prove that for every number k each countable infinite group $G$ admits a partition $G=A\cup B$ into two sets which are $k$-meager in the sense that for every $k$-element subset $K\subset G$ the sets $KA$ and $KB$ are not thick. The proof…

A group $G$ is called hereditarily non-topologizable if, for every $H\le G$, no quotient of $H$ admits a non-discrete Hausdorff topology. We construct first examples of infinite hereditarily non-topologizable groups. This allows us to prove…

It is well known that a dense subgroup $G$ of the complex unitary group $U(d)$ cannot be amenable as a discrete group when $d>1$. When $d$ is large enough we give quantitative versions of this phenomenon in connection with certain estimates…

We show that if $M$ is a compact oriented surface of genus 0 and $G$ is a subgroup of $\Symp^\omega_\mu(M)$ which has an infinite normal solvable subgroup, then $G$ is virtually abelian. In particular the centralizer of an infinite order $f…