Related papers: Permutations that Destroy Arithmetic Progressions …

We study the fundamental group of the $p$-subgroup complex of a finite group $G$. We show first that $\pi_1(A_3(A_{10}))$ is not a free group (here $A_{10}$ is the alternating group on $10$ letters). This is the first concrete example in…

Let $A$ be a ring with $1\neq 0$, not necessarily finite, endowed with an involution~$*$, that is, an anti-automorphism of order $\leq 2$. Let $H_n(A)$ be the additive group of all $n\times n$ hermitian matrices over $A$ relative to $*$.…

The introduction of a non-abelian gauge group embedded into the rigid symmetry group G of a field theory with abelian vector fields and no corresponding charges, requires in general the presence of a hierarchy of p-form gauge fields. The…

A permutation of the positive integers avoiding monotone arithmetic progressions of length $4$ with odd common difference was constructed in (LeSaulnier and Vijay, 2011). We generalise this result and show that for each $k\geq 1$, there…

Let $G$ be a finite group and $p\in \pi(G)$, and let Irr$(G)$ be the set of all irreducible complex characters of $G$. Let $\chi \in {\rm Irr}(G)$, we write ${\rm cod}(\chi)=|G:{\rm ker} \chi|/\chi(1)$, and called it the codegree of the…

For any (Hausdorff) compact group $G$ with the normalized Haar measure ${\mathbf m}_G$, denote by ${\rm cp}(G)$ the probability ${\mathbf m}_{G\times G}(\{(x,y)\in G\times G \;|\; xy=yx\})$ of commuting a randomly chosen pair of elements of…

Let G be a group of permutations of a denumerable set E. The profile of G is the function phi which counts, for each n, the number phi(n) of orbits of G acting on the n-subsets of E. Counting functions arising this way, and their associated…

Assume $G$ is a solvable group whose elementary abelian sections are all finite. Suppose, further, that $p$ is a prime such that $G$ fails to contain any subgroups isomorphic to $C_{p^\infty}$. We show that if $G$ is nilpotent, then the…

Drawing on a problem posed by Hertzsprung in 1887, we say that a given permutation $\pi\in\mathcal{S}_n$ contains the Hertzsprung pattern $\sigma\in\mathcal{S}_k$ if there is factor $\pi(d+1)\pi(d+2)\cdots\pi(d+k)$ of $\pi$ such that…

Given a countable set X (usually taken to be the natural numbers or integers), an infinite permutation, \pi, of X is a linear ordering of X. This paper investigates the combinatorial complexity of infinite permutations on the natural…

We investigate dynamical properties of the set of permutations of $\mathbb{Z}^d$ with restricted movement, i.e., permutations $\pi $ of $\mathbb{Z}^d$ such that $\pi (\mathbf{n})-\mathbf{n}$ lies, for every $\mathbf{n}\in \mathbb{Z}^d$, in…

Let $\pi, \pi'$ be irreducible tempered representations of an affine Hecke algebra H with positive parameters. We compute the higher extension groups $Ext_H^n (\pi,\pi')$ explicitly in terms of the representations of analytic R-groups…

In this paper, we finished the classification of three-generator finite $p$-groups $G$ such that $\Phi(G)\le Z(G)$. This paper is a part of classification of finite $p$-groups with a minimal non-abelian subgroup of index $p$, and partly…

Fix $a \in \mathbb{Z}$, $a\notin \{0,\pm 1\}$. A simple argument shows that for each $\epsilon > 0$, and almost all (asymptotically 100% of) primes $p$, the multiplicative order of $a$ modulo $p$ exceeds $p^{\frac12-\epsilon}$. It is an…

In this paper, we prove that every PP is an AGW-PP. We also extend the result of Wan and Lidl to other permutation polynomials over finite fields and determine their group structure. Moreover, we provide a new general method to find the…

We construct here the first known examples of non-split sharply 2-transitive groups of bounded exponent in odd positive characteristic for every large enough prime $p \equiv 3 \pmod{4}$. In fact, we show that there are countably many…

Let $A$ be a unital simple separable exact C$^*$-algebra which is approximately divisible and of real rank zero. We prove that the set of positive elements in $A$ with a fixed non-compact Cuntz class has vanishing homotopy groups. Combined…

Recall that an excedance of a permutation $\pi$ is any position $i$ such that $\pi_i > i$. Inspired by the work of Hopkins, McConville and Propp (Elec. J. Comb., 2017) on sorting using toppling, we say that a permutation is toppleable if it…

Let $n$ be a nonnegative integer. The $n$-th Ap\'{e}ry number is defined by $$ A_n:=\sum_{k=0}^n\binom{n+k}{k}^2\binom{n}{k}^2. $$ Z.-W. Sun ever investigated the congruence properties of Ap\'{e}ry numbers and posed some conjectures. For…

In the Permutation Constraint Satisfaction Problem (Permutation CSP) we are given a set of variables $V$ and a set of constraints C, in which constraints are tuples of elements of V. The goal is to find a total ordering of the variables,…