Related papers: On the polynomial automorphisms of a group

We prove a few results about non-nilpotent graphs of symmetric groups $S_n$ -- namely that they have a Hamiltonian cycle and they satisfy a conjecture of Nongsiang and Saikia. The latter is likewise proven for alternating groups $A_n$. We…

We study the class of groups having the property that every non-nilpotent subgroup is equal to its normalizer. These groups are either soluble or perfect. We completely describe the structure of soluble groups and finite perfect groups with…

Suppose that a finite group $G$ admits a Frobenius group of automorphisms $FH$ with kernel $F$ and complement $H$ such that the fixed-point subgroup of $F$ is trivial: $C_G(F)=1$. In this situation various properties of $G$ are shown to be…

In this paper, we present an explicit structure for the Baer invariant of a free nilpotent group (the $n$-th nilpotent product of the infinite cyclic group, $\textbf{Z}\st{n}* \textbf{Z}\st{n}*... \st{n}*\textbf{Z}$) with respect to the…

For any code loop $L$, we prove that the half-automorphism group of $L$ is the product of the automorphism group of $L$ by an elementary abelian $2-$group consisting of all half-automorphisms that acts as the identity on a fixed basis.…

We describe the full automorphism group of the power (di)graph of a finite group. As an application, we solve a conjecture proposed by Doostabadi, Erfanian and Jafarzadeh in 2013.

For each 3-dimensional non-Lie Leibniz algebra over the complex numbers, we describe the algebra of polynomial invariants and determine its group of automorphisms. As a consequence, we establish that any two non-nilpotent 3-dimensional…

We prove that the growth rate of an endomorphism of a finitely generated nilpotent group equals to the growth rate of induced endomorphism on its abelinization, generalizing the corresponding result for an automorphism in [14]. We also…

We show that every finitely generated nilpotent group of class 2 occurs as the quotient of a finitely presented abelian-by-nilpotent group by its largest nilpotent normal subgroup.

We show that the subgroup generated by locally finite polynomial automorphisms of k^n is normal in GA_n. Also, some properties of normal subgroups of GA_n containing all diagonal automorphisms are given.

A variety of groups does not contain all metabelian groups if and only if there is an absolute bound for the nilpotency classes of powerful $p$-groups in the given variety. Similarly, a variety contains only finitely many finite $p$-groups…

We prove that a map onto a nilpotent group $Q$ has finitely generated kernel if and only if the preimage of the positive cone is coarsely connected as a subset of the Cayley graph for every full archimedean partial order on $Q$. In case $Q$…

We find the automorphism group of the moduli space of parabolic bundles on a smooth curve (with fixed determinant and system of weights). This group is generated by: automorphisms of the marked curve, tensoring with a line bundle, taking…

Bergman has given the following abstract characterisation of the inner automorphisms of a group $G$: they are exactly those automorphisms of $G$ which can be extended functorially along any homomorphism $G \rightarrow H$ to an automorphism…

Given a finite group $G$, we denote by $\nu(G)$ the probability that two randomly chosen elements of $G$ generate a nilpotent subgroup. We prove that if $\nu(G)>1/12,$ then $G$ is solvable.

We show that the holomorph of a cyclic group of order $n$ is isomorphic to its own automophism group when $n$ is twice of a power of an odd prime.

Let A be an associative algebra of arbitrary dimension over a field F and G a finite soluble group of automorphisms of A oforder n, prime to the characteristic of F. We prove that if the fixed-point subalgebra of A under the action of G…

We define a subgroup of the universal sofic group, obtained as the normaliser of a separable abelian subalgebra. This subgroup can be obtained as an extension by the group of automorphisms on a standard probability space. We show that each…

Assume $G$ is a finite $p$-group, and let $S$ be a Sylow $p$-subgroup of $\operatorname{Aut}(G)$ with $\exp(S)=q$. We prove that if $G$ is of class $c$, then $\exp(G)|p^{\ceil{\log_pc}}q^3$, and if $G$ is a metabelian $p$-group of class at…

Cayley's theorem tells us that all groups $\mathbf{G}$ occur as subgroups of the group of automorphisms over some set $X$. In this paper we consider a `sort-of' converse to this question: given a set $X$ and some transformation group…