In this paper, we study the structure of polycyclic groups admitting an automorphism of order four on the basis of Neumann’s result, and prove that if $\alpha$ is an automorphism of order four of a polycyclic group $G$ and the map $\varphi :G\to G$ defined by $g^{\varphi }=[g,\alpha ]$ is surjective, then $G$ contains a characteristic subgroup $H$ of finite index such that the second derived subgroup $H^{\text{'}\text{'}}$ is included in the centre of $H$ and $C_H\left({\alpha }^2\right)$ is abelian, both $C_G\left({\alpha }^2\right)$ and $G/[G,{\alpha }^2]$ are abelian-by-finite. These results extend recent and classical results in...

We find representations for the automorphisms, derivations and multipliers of the Fréchet algebra $L{¹}_{loc}$ of locally integrable functions on the half-line ${ℝ}^{+}$. We show, among other things, that every automorphism θ of $L{¹}_{loc}$ is of the form $\theta ={\phi }_a{e}^{\lambda X}{e}^D$, where D is a derivation, X is the operator of multiplication by coordinate, λ is a complex number, a > 0, and ${\phi }_a$ is the dilation operator $\left({\phi }_{a}f\right)\left(x\right)=af\left(ax\right)$ ($f\in L{¹}_{loc}$, $x\in {ℝ}^{+}$). It is also shown that the automorphism group is a topological group with the topology of uniform convergence...

An automorphism a of a group G is called an artinian automorphism if for every strictly descending chain $H_1>H_2>\mathrm{\cdots }>H_n>\mathrm{\cdots }$ of subgroups of G there exists a positive integer $m$ such that ${(H_n)}^a=H_n$ for every $n\ge m$. In this paper we show that in many cases the group of all artinian automorphisms of $G$ coincides with the group of all power automorphisms of $G$.

We investigate Banach space automorphisms $T:{\ell }_{\infty }/c₀\to {\ell }_{\infty }/c₀$ focusing on the possibility of representing their fragments of the form $T_{B,A}:{\ell }_{\infty }\left(A\right)/c₀\left(A\right)\to {\ell }_{\infty }\left(B\right)/c₀\left(B\right)$ for A,B ⊆ ℕ infinite by means of linear operators from ${\ell }_{\infty }\left(A\right)$ into ${\ell }_{\infty }\left(B\right)$, infinite A×B-matrices, continuous maps from B* = βB∖B into A*, or bijections from B to A. This leads to the analysis of general bounded linear operators on ${\ell }_{\infty }/c₀$. We present many examples, introduce and investigate several classes of operators, for some of them we obtain satisfactory representations and for...

We study the reflexivity of the automorphism (and the isometry) group of the Banach algebras $L_{\infty }\left(\mu \right)$ for various measures μ. We prove that if μ is a non-atomic σ-finite measure, then the automorphism group (or the isometry group) of $L_{\infty }\left(\mu \right)$ is [algebraically] reflexive if and only if $L_{\infty }\left(\mu \right)$ is *-isomorphic to $L_{\infty }[0,1]$. For purely atomic measures, we show that the group of automorphisms (or isometries) of ${\ell }_{\infty }\left(\Gamma \right)$ is reflexive if and only if Γ has non-measurable cardinal. So, for most “practical” purposes, the automorphism...

We study conditions on automorphisms of Boolean algebras of the form $\left(\lambda \right)/{ℐ}_{\kappa }$ (where λ is an uncountable cardinal and ${ℐ}_{\kappa }$ is the ideal of sets of cardinality less than κ ) which allow one to conclude that a given automorphism is trivial. We show (among other things) that every automorphism of $\left(2^{\kappa }\right)/{ℐ}_{\kappa ⁺}$ which is trivial on all sets of cardinality κ⁺ is trivial, and that $M{A}_{\aleph ₁}$ implies both that every automorphism of (ℝ)/Fin is trivial on a cocountable set and that every automorphism of (ℝ)/Ctble is trivial. ...

Let $R$ be a prime ring of characteristic different from 2, $Q_r$ its right Martindale quotient ring and $C$ its extended centroid. Suppose that $F$, $G$ are generalized skew derivations of $R$ with the same associated automorphism $\alpha$, and $p(x_1,...,x_n)$ is a non-central polynomial over $C$ such that $$\left[F\right(x),\alpha (y\left)\right]=G\left(\right[x,y\left]\right)$$ for all $x,y\in \{p(r_1,...,r_n):r_1,...,r_n\in R\}$. Then there exists $\lambda \in C$ such that $F\left(x\right)=G\left(x\right)=\lambda \alpha \left(x\right)$ for all $x\in R$.

Let $r\left({𝔴}_2^0\right)$ be the Green ring of the weak Hopf algebra ${𝔴}_2^0$ corresponding to Sweedler’s 4-dimensional Hopf algebra $H_2$, and let $\mathrm{Aut}\left(R\left({𝔴}_2^0\right)\right)$ be the automorphism group of the Green algebra $R\left({𝔴}_2^0\right)=r\left({𝔴}_2^0\right){\otimes }_{\mathbb{Z}}ℂ$. We show that the quotient group $\mathrm{Aut}\left(R\left({𝔴}_2^0\right)\right)/C_2\cong S_3$, where $C_2$ contains the identity map and is isomorphic to the infinite group $({ℂ}^{*},\times )$ and $S_3$ is the symmetric group of order 6.

The paper was motivated by Kovacs’ paper (1973), Isaacs’ paper (1980) and a recent paper, due to Brešar et al. (2018), concerning Skolem-Noether algebras. Let $K$ be a unital commutative ring, not necessarily a field. Given a unital $K$-algebra $S$, where $K$ is contained in the center of $S$, $n\in \mathbb{N}$, the goal of this paper is to study the question: when can a homomorphism $\phi :{\mathrm{M}}_n\left(K\right)\to {\mathrm{M}}_n\left(S\right)$ be extended to an inner automorphism of ${\mathrm{M}}_n\left(S\right)$? As an application of main results presented in the paper, it is proved that if $S$ is...

Let $R$ be a noncommutative prime ring of characteristic different from 2, with its two-sided Martindale quotient ring $Q$, $C$ the extended centroid of $R$ and $a\in R$. Suppose that $\delta$ is a nonzero $\sigma$-derivation of $R$ such that $a{[\delta \left(x^n\right),x^n]}_k=0$ for all $x\in R$, where $\sigma$ is an automorphism of $R$, $n$ and $k$ are fixed positive integers. Then $a=0$.

A metacyclic group $H$ can be presented as $\langle \alpha ,\beta :{\alpha }^n=1$, ${\beta }^m={\alpha }^t$, $\beta \alpha {\beta }^{-1}={\alpha }^r\rangle$ for some $n$, $m$, $t$, $r$. Each endomorphism $\sigma$ of $H$ is determined by $\sigma \left(\alpha \right)={\alpha }^{x_1}{\beta }^{y_1}$, $\sigma \left(\beta \right)={\alpha }^{x_2}{\beta }^{y_2}$ for some integers $x_1$, $x_2$, $y_1$, $y_2$. We give sufficient and necessary conditions on $x_1$, $x_2$, $y_1$, $y_2$ for $\sigma$ to be an automorphism.

Let $f$ be a non-invertible holomorphic endomorphism of a projective space and $f^n$ its iterate of order $n$. We prove that the pull-back by $f^n$ of a generic (in the Zariski sense) hypersurface, properly normalized, converges to the Green current associated to $f$ when $n$ tends to infinity. We also give an analogous result for the pull-back of positive closed $(1,1)$-currents and a similar result for regular polynomial automorphisms of ${ℂ}^k$.

Let $G$ be a group generated by a set of finite order elements. We prove that any bicrossed product $H_{m,d}\left(q\right)\bowtie k\left[G\right]$ between the generalized Taft algebra $H_{m,d}\left(q\right)$ and group algebra $k\left[G\right]$ is actually the smash product $H_{m,d}\left(q\right)♯k\left[G\right]$. Then we show that the classification of these smash products could be reduced to the description of the group automorphisms of $G$. As an application, the classification of $H_{m,d}\left(q\right)\bowtie k[C_{n_1}\times C_{n_2}]$ is completely presented by generators and relations, where $C_n$ denotes the $n$-cyclic group.

Suppose that $G$ is a finite group and $H$ is a subgroup of $G$. Subgroup $H$ is said to be weakly $ℳ$-supplemented in $G$ if there exists a subgroup $B$ of $G$ such that (1) $G=HB$, and (2) if $H_1/H_G$ is a maximal subgroup of $H/H_G$, then $H_{1}B=B{H}_1<G$, where $H_G$ is the largest normal subgroup of $G$ contained in $H$. We fix in every noncyclic Sylow subgroup $P$ of $G$ a subgroup $D$ satisfying $1<\left|D\right|<\left|P\right|$ and study the $p$-nilpotency of $G$ under the assumption that every subgroup $H$ of $P$ with $\left|H\right|=\left|D\right|$ is weakly $ℳ$-supplemented in $G$. Some recent results are generalized. ...

A subgroup $H$ of a finite group $G$ is said to be $ss$-supplemented in $G$ if there exists a subgroup $K$ of $G$ such that $G=HK$ and $H\cap K$ is $s$-permutable in $K$. In this paper, we first give an example to show that the conjecture in A. A. Heliel’s paper (2014) has negative solutions. Next, we prove that a finite group $G$ is solvable if every subgroup of odd prime order of $G$ is $ss$-supplemented in $G$, and that $G$ is solvable if and only if every Sylow subgroup of odd order of $G$ is $ss$-supplemented in $G$. These results...