Related papers: Classification of continuously transitive circle g…

We study the closures of subgroups, semilattices and different kinds of semigroup extensions in semitopological inverse semigroups with continuous inversion. In particularly we show that a topological group $G$ is $H$-closed in the class of…

Given two subgroups $H,K$ of a compact group $G$, the probability that a random element of $H$ commutes with a random element of $K$ is denoted by $Pr(H,K)$. We show that if $G$ is a profinite group containing a Sylow $2$-subgroup $P$, a…

We give a classification of the $p$--local stable homotopy type of $BG$, where $G$ is a finite group, in purely algebraic terms. $BG$ is determined by conjugacy classes of homomorphisms from $p$--groups into $G$. This classification greatly…

Given two subsets $X,Y$ of a finite group $G$, we write $\Pr(X,Y)$ for the probability that random elements $x \in X$ and $y \in Y$ commute. If $X,Y$ are subgroups, we denote by $\Pr^*(X,Y)$ the maximum real number $\epsilon$ with the…

Here is a simplified proof that every sharply transitive subset of $\mathrm{PGL}_2(K)$ is a coset of a subgroup.

In this paper, we consider the continuity of the inverse in (strongly) paratopological gyrogroups. The conclusions are established as follows: (1) A compact Hausdorff paratopological gyrogroup $G$ is a topological gyrogroup. (2) A Hausdorff…

A subgroup of $\mathrm{Homeo}_+(\mathbb{S}^1)$ is M\"obius-like if every element is conjugate to an element of $\mathrm{PSL}(2,\mathbb{R})$. In general, a M\"obius-like subgroup of $\mathrm{Homeo}_+(\mathbb{S}^1)$ is not necessarily…

We construct a finitely presented, infinite, simple group that acts by homeomorphisms on the circle, but does not admit a non-trivial action by $C^1$-diffeomorphisms on the circle. The group emerges as a group of piecewise projective…

We study a simple problem that arises from the study of Lorentz surfaces and Anosov flows. For a non decreasing map of degree one $h:\mathbb{S}^1\to \mathbb{S}^1$, we are interested in groups of circle diffeomorphisms that act on the…

A group $G$ is self-similar if it admits a triple $(G,H,f)$ where $H$ is a subgroup of $G$ and $f: H \to G$ a simple homomorphism, that is, the only subgroup $K$ of $H$, normal in $G$ and $f$-invariant ($K^f \leq K$) is trivial. The group…

Let $G$ be a finite group and $p^k$ be a prime power dividing $|G|$. A subgroup $H$ of $G$ is called to be $\mathcal{M}$-supplemented in $G$ if there exists a subgroup $K$ of $G$ such that $G=HK$ and $H_iK<G$ for every maximal subgroup…

For G = SL(3,R) and G = SO(2,n), we give explicit, practical conditions that determine whether or not a closed, connected subgroup H of G has the property that there exists a compact subset C of G with CHC = G. To do this, we fix a Cartan…

We classify all finite 2-groups that have a cyclic or dihedral maximal subgroup and determine their automorphism groups. Based on this result, we classify all pairs $ (G,\mathcal{M}) $, such that $ G $ is a finite 2-group and $ \mathcal{M}…

Let $F$ be any field. We give a short and elementary proof that any finite subgroup $G$ of $PGL(2,F)$ occurs as a Galois group over the function field $F(x)$. We also develop a theory of descent to subfields of $F$. This enables us to…

Suppose that a group $G$ acts transitively on the points of a non-Desarguesian plane, $\mathcal{P}$. We prove first that the Sylow 2-subgroups of $G$ are cyclic or generalized quaternion. We also prove that $\mathcal{P}$ must admit an odd…

Let G be a lattice in PSL(2,C). The pro-normal topology on G is defined by taking all cosets of non-trivial normal subgroups as a basis. This topology is finer than the pro-finite topology, but it is not discrete. We prove that every…

We give a complete characterization of abelian subgroups of GL(n, R) with a locally dense (resp. dense) orbit in R^n. For finitely generated subgroups, this characterization is explicit and it is used to show that no abelian subgroup of…

Let $M$ be a closed surface. By $\Homeo(M)$ we denote the group of orientation preserving homeomorphisms of $M$ and let $\MC(M)$ denote the Mapping class group. In this paper we complete the proof of the conjecture of Thurston that says…

We establish the existence of maximal subgroups of various diferent natures in SL(n,Z). In particular, we prove that there are continuously many maximal subgroups, we provide a maximal subgroup whose action on the projective space has no…

Let G be a locally compact, Hausdorff groupoid in which s is a local homeomorphism and the unit space is totally disconnected. Assume there is a continuous cocycle c from G into a discrete group $\Gamma$. We show that the collection A(G) of…