Related papers: Permutations that Destroy Arithmetic Progressions …

Let $G$ be a finite group of order divisible by a prime $p$ and let $P\in\Syl_p(G)$. We prove a recent conjecture by Hung stating that $|\Irr_{p'}(G)|\geq \frac{\exp(P/P')-1}{p-1}+2\sqrt{p-1}-1.$ Let $a\geq 2$ be an integer and suppose that…

We prove that if $G$ is a transitive permutation group of sufficiently large degree $n$, then either $G$ is primitive and Frobenius, or the proportion of derangements in $G$ is larger than $1/(2n^{1/2})$. This is sharp, generalizes…

We show that for any permutation $\pi$ there exists an integer $k_{\pi}$ such that every permutation avoiding $\pi$ as a pattern is a product of at most $k_{\pi}$ separable permutations. In other words, every strict class $\mathcal C$ of…

We prove that for an abelian group $G$ of order $n$ the morphism $ \varphi\colon \mathbf{P}(V^*)\longrightarrow \mathbf{P} ((\mathrm{sym}^n V^*)^G)$ defined by $\varphi([f]) = [\prod_{\sigma\in G} \sigma \cdot f ]$ is nondegenerate for…

In this paper, we confirm several conjectures posed by Sun recently; for example, we prove that for any odd prime $p$ we have $$ \sum_{k=0}^{p-1}A_k\equiv\begin{cases}4x^2-2p\pmod{p^2}\quad&\text{if $p=x^2+2y^2\ (x,y\in\mathbb{Z})$},\\…

Let $p$ be an odd prime. Define the Gaussian power sum \[ G_n(p)=\sum_{a=1}^{p-1}\sum_{b=1}^{p-1}(a+bi)^n\in\mathbb Z[i]. \] We determine $G_p(p)$ modulo high powers of $p$: if $p\equiv 1\pmod 4$ then $$G_p(p)\equiv p^2(1+i)\pmod{p^3},$$…

We prove that there is an absolute constant $c>0$ with the following property: if $Z/pZ$ denotes the group of prime order $p$, and a subset $A\subset Z/pZ$ satisfies $1<|A|<p/2$, then for any positive integer…

Let $G$ be a nontrivial permutation group of degree $n$. If $G$ is transitive, then a theorem of Jordan states that $G$ has a derangement. Equivalently, a finite group is never the union of conjugates of a proper subgroup. If $G$ is…

Let $G$ be a commutative algebraic group defined over a number field $K$ that is disjoint over $K$ to $\mathbb G_a$ and satisfies the condition of semistability. Consider a linear form $l$ on the Lie algebra of $G$ with algebraic…

Let $1<a<b$ be two relatively prime integers and $\mathbb{Z}_{\ge 0}$ the set of non-negative integers. For any non-negative integer $\ell$, denote by $g_{\ell,a,b}$ the largest integer $n$ such that the equation $$n=ax+by,\quad…

We obtain an asymptotic upper bound for the product of the $p$-parts of the orders of certain composition factors of a finite group acting completely reducibly and faithfully on a finite vector space of order divisible by a prime $p$. An…

We prove a conjecture of Peter Neumann from 1966, predicting that every finite non-regular primitive permutation group of degree $n$ contains an element fixing at least one point and at most $n^{1/2}$ points. In fact, we prove a stronger…

A permutation $\pi \in \mathbb{S}_n$ is $k$-balanced if every permutation of order $k$ occurs in $\pi$ equally often, through order-isomorphism. In this paper, we explicitly construct $k$-balanced permutations for $k \le 3$, and every $n$…

Recent investigations on the set of commutators between the elements of a finite group having relatively prime orders have prompt us to propose a variant of the Ore conjecture: For every finite non-abelian simple group and for every $g\in…

Let $G \leqslant {\rm Sym}(\Omega)$ be a finite transitive permutation group and recall that an element in $G$ is a derangement if it has no fixed points on $\Omega$. Let $\Delta(G)$ be the set of derangements in $G$ and define $\delta(G) =…

In this note, we prove: \medskip \noindent {\bf Theorem A:} \emph{ There is a fixed constant $C$ such that for any positive integer $n$ and prime $p$, every finite subgroup $G$ of order coprime to $p$ of ${\rm GL}(n,\mathbb{C})$ has an…

The spherical principal series representations $\pi(\nu)$ of SL(2,$\mathbb R$) is a family of infinite dimensional representations parametrized by $\nu\in\mathbb C$. The representation $\pi(\nu)$ is irreducible unless $\nu$ is an odd…

A triangle group is denoted by $\Delta(p,q,r)$ and has finite presentation $$ \Delta(p,q,r)=\langle x,y | x^p=y^q=(xy)^r=1 \rangle .$$ We examine a method for composition of permutation representations of a triangle group $\Delta(p,q,r)$…

Let $G$ be a complex reductive group and $H=G^{\theta}$ be its fixed point subgroup under a Galois involution $\theta$. We show that any $H$-distinguished representation $\pi$ (i.e $\mathrm{dim}_{\mathbb{C}}\left(\pi^{*}\right)^{H}\neq0$)…

The defining characteristic of an exceptional point (EP) in the parameter space of a family of operators is that upon encircling the EP eigenstates are permuted. In case one encircles multiple EPs, the question arises how to properly…