Related papers: Automorphisms of Partially Commutative Groups I: L…

The main results in this thesis deal with the representation growth of certain classes of groups. In chapter $1$ we present the required preliminary theory. In chapter $2$ we introduce the Congruence Subgroup Problem for an algebraic group…

This work is a continuation of Automorphisms of $K$-groups I, P. Flavell, preprint. The main object of study is a finite $K$-group $G$ that admits an elementary abelian group $A$ acting coprimely. For certain group theoretic properties…

This is a survey of results on partially commutative groups and partially commutative algebras.

Let G be the group of points of a quasi-split reductive algebraic group over a local field F. It follows from the local Langlands conjectures that to every non-trivial additive character of F and every representation of the Langlands dual…

Let $G\subset GL_n(k)$ be a finite subgroup and $k[x_1,\dots, x_n]^G\subset k[x_1,\dots, x_n]$ its ring of invariants. We show that, in many cases, the automorphism group of $k[x_1,\dots, x_n]^G$ is $k^\times$. Version 2: Incorporates parts…

We prove that finitely generated purely loxodromic subgroups of a right-angled Artin group $A(\Gamma)$ fulfill equivalent conditions that parallel characterizations of convex cocompactness in mapping class groups $\text{Mod}(S)$. In…

The purpose of this note is to define a graph whose vertex set is a finite group $G$, whose edge set is contained in that of the commuting graph of $G$ and contains the enhanced power graph of $G$. We call this graph the deep commuting…

A map is a connected topological graph $\Gamma$ cellularly embedded in a surface. In this paper, applying Tutte's algebraic representation of map, new ideas for enumerating non-equivalent orientable or non-orientable maps of graph are…

Let $G$ be a graph with vertex set $V(G)$, $f$ a permutation of $V(G)$. Define $\delta_f(G)=|d(x,y)-d(f(x),f(y))|$ and $\delta_f(G)=\Sigma\delta_f(x,y)$, where the sum is taken over all unordered pair $x$, $y$ of distinct vertices of $G$.…

Consider a smooth connected algebraic group $G$ acting on a normal projective variety $X$ with an open dense orbit. We show that Aut($X$) is a linear algebraic group if so is $G$; for an arbitrary $G$, the group of components of Aut($X$) is…

A countable graph is ultrahomogeneous if every isomorphism between finite induced subgraphs can be extended to an automorphism. Woodrow and Lachlan showed that there are essentially four types of such countably infinite graphs: the random…

The intersection ideal graph $\Gamma(S)$ of a semigroup $S$ is a simple undirected graph whose vertices are all nontrivial left ideals of $S$ and two distinct left ideals $I, J$ are adjacent if and only if their intersection is nontrivial.…

We study when a group of form $G\times\mathbb{Z}^m (m\geq 1)$ has the finitely generated fixed subgroup property of automorphisms ($\rm{FGFP}_a$), by using the BNS-invariant, and provide some partial answers and non-trivial examples.

Let $G$ be a finite group and $\sigma$ a partition of the set of all? primes $\Bbb{P}$, that is, $\sigma =\{\sigma_i \mid i\in I \}$, where $\Bbb{P}=\bigcup_{i\in I} \sigma_i$ and $\sigma_i\cap \sigma_j= \emptyset $ for all $i\ne j$. If $n$…

Let G be a finitely generated relatively hyperbolic group. We show that if no peripheral subgroup of G is hyperbolic relative to a collection of proper subgroups, then the fixed subgroup of every automorphism of G is relatively quasiconvex.…

In this short note we prove that a graph product $G_\Gamma$ of finitely generated abelian groups is semicomplete -- that is the kernel of the natural homomorphism ${\rm Aut}(G_\Gamma)\to{\rm Aut}(G_\Gamma^{ab})$ induced by the…

The power graph of a group is the graph whose vertex set is the set of nontrivial elements of group, two elements being adjacent if one is a power of the other. We introduce some way for find the automorphism groups of some graphs. As an…

Many common finite p-groups admit automorphisms of order coprime to p, and when p is odd, it is reasonably difficult to find finite p-groups whose automorphism group is a p-group. Yet the goal of this paper is to prove that the automorphism…

The cyclic graph $\Gamma(S)$ of a semigroup $S$ is the simple graph whose vertex set is $S$ and two vertices $x, y$ are adjacent if the subsemigroup generated by $x$ and $y$ is monogenic. In this paper, we classify the semigroup $S$ such…

We compute the group of automorphisms of an arbitrary ind-variety of (possibly isotropic) generalized flags. Such an ind-variety is a homogeneous ind-space for one of the ind-groups $SL(\infty)$, $O(\infty)$ or $Sp(\infty)$. We show that…