Related papers: On the acceptable elements

A cyclic subgroup $N$ of a finite group $G$ is called a uni-width subgroup of $G$ if $N$ is the unique cyclic subgroup of $G$ of order $|N|$. In this article, we prove that a finite group $G$ admits a unique largest uni-width subgroup…

We study the Iwasawa-type decomposition of an open subset of SL(n,C) as SU(p,q)AN. We show that the dressing action of SU(p,q) is globally defined on the space of admissible elements in AN. We also show that the space of admissible elements…

For $g \geq 2$ and $h \geq 3$, we give small improvements on the maximum size of a $B_h[g]$-set contained in the interval $\{1,2, \dots , N \}$. In particular, we show that a $B_3[g]$-set in $\{1,2, \dots , N \}$ has at most $(14.3 g…

Let k be an algebraically closed field of positive characteristic and G a simple algebraic group defined over k. Under the assumption that the characteristic is a good prime for G, we determine a maximal G-stable subvariety U' of the…

Let $G$ be a $p$-group for some prime $p$. Let $n$ be the positive integer so that $|G:Z(G)| = p^n$. Suppose $A$ is a maximal abelian subgroup of $G$. Let $$p^l = {\rm max} \{|Z(C_G (g)):Z(G)| : g \in G \setminus Z(G)\},$$ $$p^b = {\rm max}…

We classify irreducible unitary representations of the group of all infinite matrices over a $p$-adic field ($p\ne 2$) with integer elements equipped with a natural topology. Any irreducible representation passes through a group $GL$ of…

Let $G$ be a simple exceptional algebraic group of adjoint type over an algebraically closed field of characteristic $p>0$ and let $X = {\rm PSL}_{2}(p)$ be a subgroup of $G$ containing a regular unipotent element $x$ of $G$. By a theorem…

Let $G$ be the unramified unitary group $U(2, 1)(E/F)$ over a non-archimedean local field $F$ of odd residue characteristic $p$. In this paper, for any supersingular representation of $G$ that contains the Steinberg weight, we prove its…

Let $G$ be a torsionfree compact $p$-adic analytic group. We give sufficient conditions on $p$ and $G$ which ensure that the Iwasawa algebra $\Omega_G$ of $G$ has no non-trivial two-sided reflexive ideals. Consequently, these conditions…

The lack of a $p$-adic Haar measure causes many methods of traditional representation theory to break down when applied to continuous representations of a compact $p$-adic Lie group $G$ in Banach spaces over a given $p$-adic field $K$. For…

Let G be a semisimple almost simple algebraic group defined and split over a nonarchimedean local field K and let V be a unipotent representation of G(K) (for example, an Iwahori-spherical representation). We calculate the character of V at…

We show that various properties of unipotent elements in a reductive group over the complex numbers can be recovered purely in terms of the affine Weyl group of the dual group.

Suppose that $A$, $B$ and $S$ are non-empty subsets of a finite abelian group $G$. Then the generalized restricted sumset $$ A\stackrel{S}+B:=\{a+b:\,a\in A,\ b\in B,\ a-b\not\in S\} $$ contains at least $$ \min\{|A|+|B|-3|S|,p(G)\} $$…

We prove several surjectivity criteria for $p$-adic representations. In particular, we classify all adjoint and simply connected group schemes $G$ over the Witt ring $W(k)$ of a finite field $k$ such that the epimorphism…

We study the decomposition of a generic element $g \in G$ of a connected reductive complex algebraic group $G$ in the form $g = N(g) B(g) \bar{u} N(g)^{-1}$ where $N: G \dashrightarrow \mathcal{N}_-$ and $B : G \dashrightarrow…

Let $A$ be a nonempty finite subset of an additive abelian group $G$. Given a nonnegative integer $h$, the $h$-fold sumset $hA$ is the set of all sums of $h$ elements of $A$, and the restricted $h$-fold sumset $h^\wedge A$ is the set of all…

Let $G=\Sp(2g,\mathbb{Z})$ be the symplectic group over the integers. Given $m\in \mathbb{N}$, it is natural to ask if there exists a non-trivial matrix $A\in G$ such that $A^{m}=I$, where $I$ is the identity matrix in $G$. In this paper,…

If H and D are two orders in a central simple algebra A with D of maximal rank and containing H, the theory of representation fields describes the set of spinor genera of orders in the genus of D representing the order H. When H is…

We study a characteristic subgroup of finitely generated groups, consisting of elements with uniform upper bound for word-lengths. For a group $G$, we denote this subgroup by $G_{bound}$. We give sufficient criteria for triviality and…

This paper is an attempt to find out which properties of a finite group G can be expressed in terms of commutators of elements of coprime orders. A criterion of solubility of G in terms of such commutators is obtained. We also conjecture…