Related papers: Automorphisms of $\kappa$-existentially closed gro…

Let S=Sym(\Omega) be the group of all permutations of a countably infinite set \Omega, and for subgroups G_1, G_2\leq S let us write G_1\approx G_2 if there exists a finite set U\subseteq S such that < G_1\cup U > = < G_2\cup U >. It is…

We describe the autotopism group Atp(G) of any abelian group G as being a semidirect product of its automorphism group Aut(G) and G^2. We then provide the subgroup structure of Atp(G) when G is a finite cyclic group.

Let G be a torsion--free abelian group of finite rank. The automorphism group Aut(G) acts on the set of maximal independent subsets of G. The orbits of this action are the isomorphism classes of indecomposable decompositions of G. G…

Throughout this Abstract, $G$ is a topological Abelian group and $\hat{G}$ is the space of continuous homomorphisms from $G$ into $T$ in the compact-open topology. A dense subgroup $D$ of $G$ determines $G$ if the (necessarily continuous)…

In \cite{MbPe}, Mbombo and Pestov prove that the group of isometries of the generalized Urysohn space of density $\kappa$, for uncountable $\kappa$ such that $\kappa^{<\kappa}=\kappa$, is not a universal topological group of weight…

Let $X$ be a compact K\"ahler manifold. We study subgroups $G \le \mathrm{Aut}(X)$ of biholomorphic automorphisms of zero entropy when $\mathrm{Aut}^0(X)$ is compact (e.g. when $\mathrm{Aut}^0(X)$ is trivial). We show that the virtual…

Let $G$ be a finite group and $\Aut(G)$ the automorphism group of $G$. The autocommuting probability of $G$, denoted by $\Pr(G, \Aut(G))$, is the probability that a randomly chosen automorphism of $G$ fixes a randomly chosen element of $G$.…

In this second part we study first the group $Aut_{\mathbb Q}(S)$ of numerically trivial automorphisms of an algebraic properly elliptic surface $S$, that is, of a minimal algebraic surface with Kodaira dimension $\kappa(S)=1$, in the case…

For a group $A$ acting by automorphisms on a group $G$, let $I_G(A)$ denote the set of commutators $[g,a]=g^{-1}g^a$, where $g\in G$ and $a\in A$, so that $[G,A]$ is the subgroup generated by $I_G(A)$. We prove that if $A$ is a $\pi$-group…

Let $K$ be a field and $f:\mathbb{P}^N \to \mathbb{P}^N$ a morphism. There is a natural conjugation action on the space of such morphisms by elements of the projective linear group $\text{PGL}_{N+1}$. The group of automorphisms, or…

Let $G$ be a group. The orbits of the natural action of $\mbox{Aut}(G)$ on $G$ are called "automorphism orbits" of $G$, and the number of automorphism orbits of $G$ is denoted by $\omega(G)$. In this paper the finite nonsolvable groups $G$…

We study the automorphism group of graphons (graph limits). We prove that after an appropriate "standardization" of the graphon, the automorphism group is compact. Furthermore, we characterize the orbits of the automorphism group on…

In this paper we study different questions concerning automorphisms of quandles. For a conjugation quandle $Q={\rm Conj}(G)$ of a group $G$ we determine several subgroups of ${\rm Aut}(Q)$ and find necessary and sufficient conditions when…

Let $G $ be a group of cardinality $\kappa>\aleph_0 $ endowed with a topology $\tau $ such that $|U|=\kappa$ for every non-empty $U\in\tau$ and $\tau$ has a base of cardinality $\kappa$. We prove that $G$ could be factorized $G=AB$ (i.e.…

Let k be an algebraically closed field of characteristic zero, F its algebraically closed extension, and G be the group of k-automorphisms of F endowed with a natural topology. One of the purposes of this paper is to show that any…

Let $R$ be a commutative $k-$algebra over a field $k$. Assume $R$ is a noetherian, infinite, integral domain. The group of $k-$automorphisms of $R$,i.e.$Aut_k(R)$ acts in a natural way on $(R-k)$.In the first part of this article, we study…

We give a complete classification of the finite $2$-groups $G$ for which the automorphism group $\operatorname{Aut}(G)$ acting naturally on $G$ has three orbits. There are two infinite families and one additional group, of order $2^9$. All…

We describe the structure of virtually solvable normal subgroups in the automorphism group of a right-angled Artin group ${\rm Aut}(A_\Gamma)$. In particular, we prove that a finite normal subgroup in ${\rm Aut}(A_\Gamma)$ has at most order…

For a finite group G of Lie type and a prime p, we compare the automorphism groups of the fusion and linking systems of G at p with the automorphism group of G itself. When p is the defining characteristic of G, they are all isomorphic,…

Let $G$ be a finite group admitting a coprime automorphism $\alpha$ of order $e$. Denote by $I_G(\alpha)$ the set of commutators $g^{-1}g^\alpha$, where $g\in G$, and by $[G,\alpha]$ the subgroup generated by $I_G(\alpha)$. We study the…