Related papers: Profinite groups with an automorphism whose fixed …

Let G be a reductive group over a commutative ring R. We say that G has isotropic rank >=n, if every normal semisimple reductive R-subgroup of G contains (G_m)^n. We prove that if G has isotropic rank >=1 and R is a regular domain…

Let R be a Lie nilpotent algebra of index k over a field K of characteristic zero. If G is an n-element subgroup of Aut(R) of the K-automorphisms, then we prove that R is right integral over Fix(G) of degree n^k. In the presence of a…

It is well known that the automorphism group of a regular dessin is a two-generator finite group, and the isomorphism classes of regular dessins with automorphism groups isomorphic to a given finite group $G$ are in one-to-one…

For any right-angled Artin group, we show that its outer automorphism group contains either a finite-index nilpotent subgroup or a nonabelian free subgroup. This is a weak Tits alternative theorem. We find a criterion on the defining graph…

A topological group $G$ has the Approximate Fixed Point (AFP) property on a bounded convex subset $C$ of a locally convex space if every continuous affine action of $G$ on $C$ admits a net $(x_i)$, $x_i\in C$, such that…

A group G is a cn-group if for each subgroup H of G there exists a normal subgroup N of G such that the index of both H and N in HN is finite. The class of cn-groups contains properly the classes of core- finite groups and that of groups in…

We give a geometric characterization of finite rational groups. In particular, we prove that a finite group is rational if and only if there exists a finite geometry $\Gamma$ of type $I$ and action of $G$ on $\Gamma$ as a group of…

A generalized Baumslag-Solitar group (GBS group) is a finitely generated group $G$ which acts on a tree with all edge and vertex stabilizers infinite cyclic. We show that Out(G) either contains non-abelian free groups or is virtually…

A finite group $G$ is said to be rational if every character of $G$ is rational-valued. The Gruenberg-Kegel graph of a finite group $G$ is the undirected graph whose vertices are the primes dividing the order of $G$ and the edges join…

A group G has restricted centralizers if for each g in G the centralizer C_G(g) either is finite or has finite index in G. A theorem of Shalev states that a profinite group with restricted centralizers is abelian-by-finite. In the present…

We introduce a directed graph related to a group $G$, which we call the N-prime graph $\Gamma_{\rm{N}}(G)$ of $G$ and which is a refinement of the classical Gruenberg-Kegel graph. The vertices of $\Gamma_{\rm{N}}(G)$ are the primes $p$ such…

Given a finite group $G,$ we denote by $\Delta(G)$ the graph whose vertices are the proper subgroups of $G$ and in which two vertices $H$ and $K$ are joined by an edge if and only if $G=\langle H,K\rangle.$ We prove that if there exists a…

We give necessary and sufficient conditions on the graph of a right-angled Artin group that determine whether the group is subgroup separable or not. Moreover, we investigate the profinite topology of the direct product of two free groups.…

A locally compact group $G$ is said to be $\ast$-regular if the natural map $\Psi:\Prim C^\ast(G)\to\Prim_{\ast} L^1(G)$ is a homeomorphism with respect to the Jacobson topologies on the primitive ideal spaces $\Prim C^\ast(G)$ and…

We prove that infinite definably simple locally finite groups of finite centraliser dimension are simple groups of Lie type over locally finite fields. Then, we identify conditions on automorphisms of a stable group that make it resemble…

Assume that a finite group $G$ admits a Frobenius group of automorphisms $FH$ with kernel $F$ and complement $H$ such that $C_{G}(F)=1$. In this paper, we investigate this situation and prove that if $C_G(H)$ is supersoluble and $C_{G'}(H)$…

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…

A profinite group $G$ is just infinite if every closed normal subgroup of $G$ is of finite index. We prove that an infinite profinite group is just infinite if and only if, for every open subgroup $H$ of $G$, there are only finitely many…

A profinite group is called small if it has only finitely many open subgroups of index n for each positive integer n. We show that every Frattini cover of a small profinite group is small. A profinite group is called strongly complete if…

An automorphism $\alpha$ of a group $G$ is called a commuting automorphism if each element $x$ in $G$ commutes with its image $\alpha(x)$ under $\alpha$. Let $A(G)$ denote the set of all commuting automorphisms of $G$. Rai [Proc. Japan…