A ring extension $R\subseteq S$ is said to be strongly affine if each $R$-subalgebra of $S$ is a finite-type $R$-algebra. In this paper, several characterizations of strongly affine extensions are given. For instance, we establish that if $R$ is a quasi-local ring of finite dimension, then $R\subseteq S$ is integrally closed and strongly affine if and only if $R\subseteq S$ is a Prüfer extension (i.e. $(R,S)$ is a normal pair). As a consequence, the equivalence of strongly affine extensions, quasi-Prüfer extensions and INC-pairs is shown....

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...

Given an integral scheme $X$ over a non-archimedean valued field $k$, we construct a universal closed embedding of $X$ into a $k$-scheme equipped with a model over the field with one element ${𝔽}_1$ (a generalization of a toric variety). An embedding into such an ambient space determines a tropicalization of $X$ by previous work of the authors, and we show that the set-theoretic tropicalization of $X$ with respect to this universal embedding is the Berkovich analytification $X^{\mathrm{an}}$. Moreover, using the scheme-theoretic...

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 $k$ be a field of characteristic $p\>0$. Let $D_m$ be a ${BT}_m$ over $k$ (i.e., an $m$-truncated Barsotti–Tate group over $k$). Let $S$ be a $k$-scheme and let $X$ be a ${BT}_m$ over $S$. Let $S_{D_m}\left(X\right)$ be the subscheme of $S$ which describes the locus where $X$ is locally for the fppf topology isomorphic to $D_m$. If $p\ge 5$, we show that $S_{D_m}\left(X\right)$ is pure in $S$, i.e. the immersion $S_{D_m}\left(X\right)\hookrightarrow S$ is affine. For $p\in \{2,3\}$, we prove purity if $D_m$ satisfies a certain technical property depending only on its $p$-torsion $D_m\left[p\right]$. For $p\ge 5$, we apply the developed techniques to show that...

Let $Q$ be a finite union of Dynkin quivers, $G\subseteq \mathrm{Aut}\left(𝕜Q\right)$ a finite abelian group, $\widehat{Q}$ the generalized McKay quiver of $(Q,G)$ and ${\Gamma }_Q$ the Auslander-Reiten quiver of $𝕜Q$. Then $G$ acts functorially on the quiver ${\Gamma }_Q$. We show that the Auslander-Reiten quiver of $𝕜\widehat{Q}$ coincides with the generalized McKay quiver of $({\Gamma }_Q,G)$.

We study here several finiteness problems concerning affine Nash manifolds $M$ and Nash subsets $X$. Three main results are: (i) A Nash function on a semialgebraic subset $Z$ of $M$ has a Nash extension to an open semialgebraic neighborhood of $Z$ in $M$, (ii) A Nash set $X$ that has only normal crossings in $M$ can be covered by finitely many open semialgebraic sets $U$ equipped with Nash diffeomorphisms $(u_1,\cdots ,u_m):U\to {ℝ}^m$ such that $U\cap X=\{u_1\cdots u_r=0\}$, (iii) Every affine Nash manifold with corners $N$ is a closed subset of an affine Nash...

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.

Let $N$ be a normal subgroup of a group $G$. The structure of $N$ is given when the $G$-conjugacy class sizes of $N$ is a set of a special kind. In fact, we give the structure of a normal subgroup $N$ under the assumption that the set of $G$-conjugacy class sizes of $N$ is $(p_{1n_1}^{a_{1n_1}},\cdots ,p_{11}^{a_{11}},1)\times \cdots \times (p_{r{n}_r}^{a_{r{n}_r}},\cdots ,p_{r1}^{a_{r1}},1)$, where $r>1$, $n_i>1$ and $p_{ij}$ are distinct primes for $i\in \{1,2,\cdots ,r\}$, $j\in \{1,2,\cdots ,n_i\}$.

Let $S$ be a fixed symmetric finite subset of $S{L}_d\left({𝒪}_K\right)$ that generates a Zariski dense subgroup of $S{L}_d\left({𝒪}_K\right)$ when we consider it as an algebraic group over $mathbbQ$ by restriction of scalars. We prove that the Cayley graphs of $S{L}_d({𝒪}_K/I)$ with respect to the projections of $S$ is an expander family if $I$ ranges over square-free ideals of ${𝒪}_K$ if $d=2$ and $K$ is an arbitrary numberfield, or if $d=3$ and $K=\mathbb{Q}$.

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...

Let $X_i$, i∈ I, and $Y_j$, j∈ J, be compact convex sets whose sets of extreme points are affinely independent and let φ be an affine homeomorphism of ${\prod }_{i\in I}{X}_i$ onto ${\prod }_{j\in J}{Y}_j$. We show that there exists a bijection b: I → J such that φ is the product of affine homeomorphisms of $X_i$ onto $Y_{b\left(i\right)}$, i∈ I.

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.

Let $G$ be a unitary group defined over a non-Archimedean local field of odd residue characteristic and let $H$ be the centralizer of a semisimple rational Lie algebra element of $G.$ We prove that the Bruhat-Tits building ${𝔅}^1\left(H\right)$ of $H$ can be affinely and $G$-equivariantly embedded in the Bruhat-Tits building ${𝔅}^1\left(G\right)$ of $G$ so that the Moy-Prasad filtrations are preserved. The latter property forces uniqueness in the following way. Let $j$ and $j^{\prime }$ be maps from ${𝔅}^1\left(H\right)$ to ${𝔅}^1\left(G\right)$ which preserve the Moy–Prasad filtrations....

Let $k$ be an algebraically closed field of characteristic $p\>0$. We study obstructions to lifting to characteristic $0$ the faithful continuous action $\phi$ of a finite group $G$ on $k\left[\right[t\left]\right]$. To each such $\phi$ a theorem of Katz and Gabber associates an action of $G$ on a smooth projective curve $Y$ over $k$. We say that the KGB obstruction of $\phi$ vanishes if $G$ acts on a smooth projective curve $X$ in characteristic $0$ in such a way that $X/H$ and $Y/H$ have the same genus for all subgroups $H\subset G$. We determine for which $G$ the KGB...

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. ...

Let $\sigma =\{{\sigma }_i:i\in I\}$ be some partition of the set of all primes $\mathbb{P}$, $G$ be a finite group and $\sigma \left(G\right)=\{{\sigma }_i:{\sigma }_i\cap \pi \left(G\right)\ne \varnothing \}$. A set $\mathscr{H}$ of subgroups of $G$ is said to be a complete Hall $\sigma$-set of $G$ if every non-identity member of $\mathscr{H}$ is a Hall ${\sigma }_i$-subgroup of $G$ and $\mathscr{H}$ contains exactly one Hall ${\sigma }_i$-subgroup of $G$ for every ${\sigma }_i\in \sigma \left(G\right)$. $G$ is said to be $\sigma$-full if $G$ possesses a complete Hall $\sigma$-set. A subgroup $H$ of $G$ is $\sigma$-permutable in $G$ if $G$ possesses a complete Hall $\sigma$-set $\mathscr{H}$ such that $H{A}^x$= $A^{x}H$ for all $A\in \mathscr{H}$ and all $x\in G$. A subgroup $H$ of $G$ is $\sigma$-permutably embedded in...