We investigate the structure of the twisted Brauer monoid , comparing and contrasting it with the structure of the (untwisted) Brauer monoid . We characterize Green's relations and pre-orders on , describe the lattice of ideals and give necessary and sufficient conditions for an ideal to be idempotent generated. We obtain formulae for the rank (smallest size of a generating set) and (where applicable) the idempotent rank (smallest size of an idempotent generating set) of each principal ideal; in particular, when an ideal is idempotent generated, its rank and idempotent rank are equal. As an application of our results, we describe the idempotent generated subsemigroup of (which is not an ideal), as well as the singular ideal of (which is neither principal nor idempotent generated), and we deduce that the singular part of the Brauer monoid is idempotent generated, a result previously proved by Maltcev and Mazorchuk.

The group of ${\mathcal{C}}^{1}$ -diffeomorphisms of any sparse Cantor subset of a manifold is countable and discrete (possibly trivial). Thompson’s groups come out of this construction when we consider central ternary Cantor subsets of an interval. Brin’s higher-dimensional generalizations $nV$ of Thompson’s group $V$ arise when we consider products of central ternary Cantor sets. We derive that the ${\mathcal{C}}^{2}$ -smooth mapping class group of a sparse Cantor sphere pair is a discrete countable group and produce this way versions of the braided Thompson groups.

We establish a new sufficient condition under which a monoid is nonfinitely based and apply this condition to Lee monoids $L_{\ell }^{1}$ , obtained by adjoining an identity element to the semigroup generated by two idempotents $a$ and $b$ with the relation $0=abab\cdots \,$ (length $\ell$ ). We show that every monoid $M$ which generates a variety containing $L_{5}^{1}$ and is contained in the variety generated by $L_{\ell }^{1}$ for some $\ell \geq 5$ is nonfinitely based. We establish this result by analysing $\unicode[STIX]{x1D70F}$ -terms for $M$ , where $\unicode[STIX]{x1D70F}$ is a certain nontrivial congruence on the free semigroup. We also show that if $\unicode[STIX]{x1D70F}$ is the trivial congruence on the free semigroup and $\ell \leq 5$ , then the $\unicode[STIX]{x1D70F}$ -terms (isoterms) for $L_{\ell }^{1}$ carry no information about the nonfinite basis property of $L_{\ell }^{1}$ .

In this paper, we introduce the Newton decomposition on a connected reductive $p$ -adic group $G$ . Based on it we give a nice decomposition of the cocenter of its Hecke algebra. Here we consider both the ordinary cocenter associated to the usual conjugation action on $G$ and the twisted cocenter arising from the theory of twisted endoscopy. We give Iwahori–Matsumoto type generators on the Newton components of the cocenter. Based on it, we prove a generalization of Howe’s conjecture on the restriction of (both ordinary and twisted) invariant distributions. Finally we give an explicit description of the structure of the rigid cocenter.

The author has previously associated to each commutative ring with unit $R$ and étale groupoid $\mathscr{G}$ with locally compact, Hausdorff and totally disconnected unit space an $R$ -algebra $R\,\mathscr{G}$ . In this paper we characterize when $R\,\mathscr{G}$ is Noetherian and when it is Artinian. As corollaries, we extend the characterization of Abrams, Aranda Pino and Siles Molina of finite-dimensional and of Noetherian Leavitt path algebras over a field to arbitrary commutative coefficient rings and we recover the characterization of Okniński of Noetherian inverse semigroup algebras and of Zelmanov of Artinian inverse semigroup algebras.

If $k$ is a positive integer, a group $G$ is said to have the $FE_{k}$ -property if for each element $g$ of $G$ there exists a normal subgroup of finite index $X(g)$ such that the subgroup $\langle g,x\rangle$ is nilpotent of class at most $k$ for all $x\in X(g)$ . Thus, $FE_{1}$ -groups are precisely those groups with finite conjugacy classes ($FC$ -groups) and the aim of this paper is to extend properties of $FC$ -groups to the case of groups with the $FE_{k}$ -property for $k>1$ . The class of $FE_{k}$ -groups contains the relevant subclass $FE_{k}^{\ast }$ , consisting of all groups $G$ for which to every element $g$ there corresponds a normal subgroup of finite index $Y(g)$ such that $\langle g,U\rangle$ is nilpotent of class at most $k$ , whenever $U$ is a nilpotent subgroup of class at most $k$ of $Y(g)$ .

Let $G$ be a finite group. Let $\operatorname{cl}(G)$ be the set of conjugacy classes of $G$ and let $\operatorname{ecl}_{p}(G)$ be the largest integer such that $p^{\operatorname{ecl}_{p}(G)}$ divides $|C|$ for some $C\in \operatorname{cl}(G)$ . We prove the following results. If $\operatorname{ecl}_{p}(G)=1$ , then $|G:F(G)|_{p}\leq p^{4}$ if $p\geq 3$ . Moreover, if $G$ is solvable, then $|G:F(G)|_{p}\leq p^{2}$ .

We show that if a finitely generated group $G$ has a nonelementary WPD action on a hyperbolic metric space $X$ , then the number of $G$ -conjugacy classes of $X$ -loxodromic elements of $G$ coming from a ball of radius $R$ in the Cayley graph of $G$ grows exponentially in $R$ . As an application we prove that for $N\geq 3$ the number of distinct $\text{Out}(F_{N})$ -conjugacy classes of fully irreducible elements $\unicode[STIX]{x1D719}$ from an $R$ -ball in the Cayley graph of $\text{Out}(F_{N})$ with $\log \unicode[STIX]{x1D706}(\unicode[STIX]{x1D719})$ of the order of $R$ grows exponentially in $R$ .

The set of row reduced matrices (and of echelon form matrices) is closed under multiplication. We show that any system of representatives for the $\text{Gl}_{n}(\mathbb{K})$ action on the $n\times n$ matrices, which is closed under multiplication, is necessarily conjugate to one that is in simultaneous echelon form. We call such closed representative systems Grassmannian semigroups. We study internal properties of such Grassmannian semigroups and show that they are algebraic semigroups and admit gradings by the finite semigroup of partial order preserving permutations, with components that are naturally in one–one correspondence with the Schubert cells of the total Grassmannian. We show that there are infinitely many isomorphism types of such semigroups in general, and two such semigroups are isomorphic exactly when they are semiconjugate in $M_{n}(\mathbb{K})$ . We also investigate their representation theory over an arbitrary field, and other connections with multiplicative structures on Grassmannians and Young diagrams.

Let $G$ be a connected linear algebraic group over a number field $k$ . Let $U{\hookrightarrow}X$ be a $G$ -equivariant open embedding of a $G$ -homogeneous space $U$ with connected stabilizers into a smooth $G$ -variety $X$ . We prove that $X$ satisfies strong approximation with Brauer–Manin condition off a set $S$ of places of $k$ under either of the following hypotheses:

1. (i) $S$ is the set of archimedean places;

2. (ii) $S$ is a non-empty finite set and $\bar{k}^{\times }=\bar{k}[X]^{\times }$.

The proof builds upon the case $X=U$ , which has been the object of several works.

Let $G$ be a group and $\unicode[STIX]{x1D70E}=\{\unicode[STIX]{x1D70E}_{i}\mid i\in I\}$ some partition of the set of all primes. A subgroup $A$ of $G$ is $\unicode[STIX]{x1D70E}$ -subnormal in $G$ if there is a subgroup chain $A=A_{0}\leq A_{1}\leq \cdots \leq A_{m}=G$ such that either $A_{i-1}\unlhd A_{i}$ or $A_{i}/(A_{i-1})_{A_{i}}$ is a finite $\unicode[STIX]{x1D70E}_{j}$ -group for some $j=j(i)$ for $i=1,\ldots ,m$ , and it is modular in $G$ if $\langle X,A\cap Z\rangle =\langle X,A\rangle \cap Z$ when $X\leq Z\leq G$ and $\langle A,Y\cap Z\rangle =\langle A,Y\rangle \cap Z$ when $Y\leq G$ and $A\leq Z\leq G$ . The group $G$ is called $\unicode[STIX]{x1D70E}$ -soluble if every chief factor $H/K$ of $G$ is a finite $\unicode[STIX]{x1D70E}_{i}$ -group for some $i=i(H/K)$ . In this paper, we describe finite $\unicode[STIX]{x1D70E}$ -soluble groups in which every $\unicode[STIX]{x1D70E}$ -subnormal subgroup is modular.

In this paper, we investigate the set of accumulation points of normalized roots of infinite Coxeter groups for certain class of their action. Concretely, we prove the conjecture proposed in [6, Section 3.2] in the case where the equipped Coxeter matrices are of type $(n-1,1)$, where $n$ is the rank. Moreover, we obtain that the set of such accumulation points coincides with the closure of the orbit of one point of normalized limit roots. In addition, in order to prove our main results, we also investigate some properties on fixed points of the action.

Let G be a finite group, and write cd (G) for the set of degrees of irreducible characters of G. The common-divisor graph Γ(G) associated with G is the graph whose vertex set is cd (G)∖{1} and there is an edge between distinct vertices a and b, if (a, b) > 1. In this paper we prove that if Γ(G) is a k-regular graph for some k ⩾ 0, then for the solvable groups, either Γ(G) is a complete graph of order k + 1 or Γ(G) has two connected components which are complete of the same order and for the non-solvable groups, either k = 0 and cd(G) = cd(PSL2(2f)), where f ⩾ 2 or Γ(G) is a 4-regular graph with six vertices and cd(G) = cd(Alt7) or cd(Sym7).

We study the germs at the origin of $G$-representation varieties and the degree 1 cohomology jump loci of fundamental groups of quasi-projective manifolds. Using the Morgan–Dupont model associated to a convenient compactification of such a manifold, we relate these germs to those of their infinitesimal counterparts, defined in terms of flat connections on those models. When the linear algebraic group $G$ is either $\text{SL}_{2}(\mathbb{C})$ or its standard Borel subgroup and the depth of the jump locus is 1, this dictionary works perfectly, allowing us to describe in this way explicit irreducible decompositions for the germs of these embedded jump loci. On the other hand, if either $G=\text{SL}_{n}(\mathbb{C})$ for some $n\geqslant 3$, or the depth is greater than 1, then certain natural inclusions of germs are strict.

The relation between the structure of a ring and the structure of its additive group is studied in the context of some recent results in additive groups of mixed rings. Namely, the notion of the square subgroup of an abelian group, which is a generalization of the concept of nil-group, is considered mainly for mixed non-splitting abelian groups which are the additive groups only of rings whose all subrings are ideals. A non-trivial construction of such a group of finite torsion-free rank no less than two, for which the quotient group modulo the square subgroup is not a nil-group, is given. In particular, a new class of abelian group for which an old problem posed by Stratton and Webb has a negative solution, is indicated. A new, far from obvious, application of rings in which the relation of being an ideal is transitive, is obtained.

Let G be a group acting freely, properly discontinuously and cellularly on some finite dimensional CW-complex Σ(2n) which has the homotopy type of the 2n-sphere 𝕊2n. Then, that action induces a homomorphism G → Aut(H2n(Σ(2n))). We classify all pairs (G, φ), where G is a virtually cyclic group and φ: G → Aut(ℤ) is a homomorphism, which are realizable in the way above and the homotopy types of all possible orbit spaces as well. Next, we consider the family of all groups which have virtual cohomological dimension one and which act on some Σ(2n). Those groups consist of free groups and semi-direct products F ⋊ ℤ2 with F a free group. For a group G from the family above and a homomorphism φ: G → Aut(ℤ), we present an algebraic criterion equivalent to the realizability of the pair (G, φ). It turns out that any realizable pair can be realized on some Σ(2n) with dim Σ(2n) ≤ 2n + 1.

Given an odd prime $p$, we investigate the position of simple modules in the stable Auslander–Reiten quiver of the principal block of a finite group with noncyclic abelian Sylow $p$-subgroups. In particular, we prove a reduction to finite simple groups. In the case that the characteristic is $3$, we prove that simple modules in the principal block all lie at the end of their components.

As a natural generalisation of $q$-Schur algebras associated with the Hecke algebra ${\mathcal{H}}_{r,R}$ (of the symmetric group), we introduce the queer $q$-Schur superalgebra associated with the Hecke–Clifford superalgebra ${\mathcal{H}}_{r,R}^{\mathsf{c}}$, which, by definition, is the endomorphism algebra of the induced ${\mathcal{H}}_{r,R}^{\mathsf{c}}$-module from certain $q$-permutation modules over ${\mathcal{H}}_{r,R}$. We will describe certain integral bases for these superalgebras in terms of matrices and will establish the base-change property for them. We will also identify the queer $q$-Schur superalgebras with the quantum queer Schur superalgebras investigated in the context of quantum queer supergroups and provide a constructible classification of their simple polynomial representations over a certain extension of the field $\mathbb{C}(\mathbf{v})$ of complex rational functions.

Let $k$ be a nonnegative integer. A subgroup $X$ of a group $G$ has normal length $k$ in $G$ if all chains between $X$ and its normal closure $X^{G}$ have length at most $k$ , and $k$ is the length of at least one of these chains. The group $G$ is said to have finite normal length if there is a finite upper bound for the normal lengths of its subgroups. The aim of this paper is to study groups of finite normal length. Among other results, it is proved that if all subgroups of a locally (soluble-by-finite) group $G$ have finite normal length in $G$ , then the commutator subgroup $G^{\prime }$ is finite and so $G$ has finite normal length. Special attention is given to the structure of groups of normal length $2$ . In particular, it is shown that finite groups with this property admit a Sylow tower.

We derive some structural properties of a trifactorised finite group $G=AB=AC=BC$ , where $A$ , $B$ , and $C$ are subgroups of $G$ , provided that $A=A_{\unicode[STIX]{x1D70B}}\times A_{\unicode[STIX]{x1D70B}^{\prime }}$ and $B=B_{\unicode[STIX]{x1D70B}}\times B_{\unicode[STIX]{x1D70B}^{\prime }}$ are $\unicode[STIX]{x1D70B}$ -decomposable groups, for a set of primes $\unicode[STIX]{x1D70B}$ .