Related papers: Finite Groups with Abelian Automorphism Groups: A …

We investigate the possible structures imposed on a finite group by its possession of an automorphism sending a large fraction of the group elements to their cubes, the philosophy being that this should force the group to be, in some sense,…

In this short survey article, we try to list maximum number of known results on class preserving automorphisms of finite $p$-groups. We conclude the survey with some interesting (at least for the author) open problems on this topic.

We construct, for the first time, various types of specific non-special finite $p$-groups having abelian automorphism group. More specifically, we construct groups $G$ with abelian automorphism group such that $\gamma_2(G) < \mathrm{Z}(G) <…

We will show that every element of a finitely generated abelian group is automorphically equivalent what we will define to be a {\em representative element} in a {\em repeat-free subgroup}, and for finite abelian groups we can count the…

Let $1 \to N \to G \to H \to 1$ be an abelian extension. The purpose of this paper is to study the problem of extending automorphisms of $N$ and lifting automorphisms of $H$ to certain automorphisms of $G$.

We survey recent results on multiple transitivity of automorphism groups of affine algebraic varieties. We consider the property of infinite transitivity of the special automorphism group, which is equivalent to flexibility of the…

A long-standing conjecture asserts that every finite non-abelian $p$-group has a non-inner automorphism of order $p$. In this paper, we settle the conjecture for a finite $p$-group ($p >2$) of nilpotency class $n$ with certain conditions.

This is a brief overview of a few selected chapters on automorphism groups of affine varieties. It includes some open questions.

The main purpose of this paper is to describe some published results and outline corresponding approaches which when applied to automorphism groups of algebras or groups establish that these groups are linear or non-linear.

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.

The target of this article is to discuss the concept of \textit{commuting probability} of finite groups which, in short, is a probabilistic measure of how abelian our group is. We shall compute the value of commuting probability for many…

In this note we survey recent results on automorphisms of affine algebraic varieties, infinitely transitive group actions and flexibility. We present related constructions and examples, and discuss geometric applications and open problems.

We characterize the groups isomorphic to full automorphism groups of ordered abelian groups. The result will follow from classical theorems on ordered groups adding an argument from proofs used to realize rings as endomorphism rings of…

We list all finite abelian groups which act effectively on smooth cubic fourfolds.

We study the automorphism group of the algebraic closure of a substructure A of a pseudo-finite field F, or more generally, of a bounded PAC field F. This paper answers some of the questions of [1], and in particular that any finite group…

We give a classification of maximal elements of the set of finite groups that can be realized as the full automorphism groups of polarized abelian surfaces over finite fields.

In this paper, we study some properties of the outer automorphism group of free Burnside groups of large odd exponent. In particular, we prove that it contains free and free abelian subgroups.

We give a classification of maximal elements of the set of finite groups that can be realized as the automorphism groups of polarized abelian threefolds over finite fields.

The main goal of this paper is to apply the arithmetic method developed in our previous paper \cite{13} to determine the number of some types of subgroups of finite abelian groups.

Recently, several works by a number of authors have provided characterizations of integral undirected Cayley graphs over generalized dihedral groups and generalized dicyclic groups. We generalize and unify these results in two different…