Related papers: On the Cycle Structure of the Metacommutation Map

In 1963, Anton Kotzig famously conjectured that $K_{n}$, the complete graph of order $n$, where $n$ is even, can be decomposed into $n-1$ perfect matchings such that every pair of these matchings forms a Hamilton cycle. The problem is still…

In this paper we study the parameterized complexity of two well-known permutation group problems which are NP-complete. 1. Given a permutation group G=<S>, subgroup of $S_n$, and a parameter $k$, find a permutation $\pi$ in G such that…

Let $p$ be an odd prime. For each integer $a$ with $p\nmid a$, the famous Zolotarev's Lemma says that the Legendre symbol $(\frac{a}{p})$ is the sign of the permutation of $\Z/p\Z$ induced by multiplication by $a$. The extension of…

The abstract of the original paper was as follows: We explore some questions related to one of Brizolis: does every prime p have a pair (g,h) such that h is a fixed point for the discrete logarithm with base g? We extend this question to…

Let $p$ be an odd prime. Denote a Sylow $p$-subgroup of $GL_2(\mathbb{Z}/p^n)$ and $SL_2(\mathbb{Z}/p^n)$ by $S_p(n,GL)$ and $S_p(n,SL)$ respectively. The theory of stable elements tells us that the mod-$p$ cohomology of a finite group is…

By the investigation of $k$-orbits symmetry properties it is obtained a simple proof of the B. Fein, W. M. Kantor and M. Schacher Theorem: any transitive permutation group contains a non-trivial fixed-point-free prime-power element. Key…

Borwein and Choi conjectured that a polynomial $P(x)$ with coefficients $\pm1$ of degree $N-1$ is cyclotomic iff $$P(x)=\pm \Phi_{p_1}(\pm x)\Phi_{p_2}(\pm x^{p_1})\cdots \Phi_{p_r}(\pm x^{p_1p_2\cdots p_{r-1}})$$ where $N=p_1p_2\cdots…

We show that for supersingular prime p the image of a unique meromorphic function G_p on X_0(p) (of degree two, with polar divisor {[0]_0,[\infty]_0}) under a certain Hecke operator is equal to j(\tau) (up to some additional constant). This…

Let $k$ be a nonnegative integer, and let $\alpha$ and $\beta$ be two permutations of $n$ symbols. We say that $\alpha$ and $\beta$ $k$-commute if $H(\alpha\beta, \beta\alpha)=k$, where $H$ denotes the Hamming metric between permutations.…

Let $r\geq 3$ be any positive integer which is relatively prime to $p$ and $q^2\equiv 1 \pmod r$. Let $\tau_1, \tau_2$ be any permutation polynomials over $\mathbb{F}_{q^2},$ $\sigma_M$ is an invertible linear map over $\mathbb{F}_{q^2}$…

Let $G$ be a finite group, let $p$ be a prime and let ${\rm Pr}_p(G)$ be the probability that two random $p$-elements of $G$ commute. In this paper we prove that ${\rm Pr}_p(G) > (p^2+p-1)/p^3$ if and only if $G$ has a normal and abelian…

Let $F$ be a $p$-adic field, that is, a finite extension of $\mathbb Q_p$. Let $D$ be a finite-dimensional central division algebra over $F$ and let $SL_1(D)$ be the group of elements of reduced norm $1$ in $D$. Prasad and Raghunathan…

We proved recently (see \cite{lhgarasu}) the result on the title for odd prime divisors of such an $n.$ The result implies for many $n's$, more precisely, for an infinity of $n$'s with an arbitrary fixed number of prime divisors, the…

In an award-winning expository article, V. Pozdnyakov and J.M. Steele gave a beautiful demonstration of the ramifications of a basic bijection for permutations. The aim of this note is to connect this correspondence to a seemingly unrelated…

This paper develops an analogy between the cycle structure of, on the one hand, random permutations with cycle lengths restricted to lie in an infinite set $S$ with asymptotic density $\sigma$ and, on the other hand, permutations selected…

Let $f(x) \in \mathbb{Z}[x]$; for each integer $\alpha$ it is interesting to consider the number of iterates $n_{\alpha}$, if possible, needed to satisfy $f^{n_{\alpha}}(\alpha) = \alpha$. The sets $\{\alpha, f(\alpha), \ldots,…

A step forward is made in a long standing Lov\'{a}sz's problem regarding hamiltonicity of vertex-transitive graphs by showing that every connected vertex-transitive graph of order a product of two primes, other than the Petersen graph,…

We prove limit theorems for the number of fixed points, descents, and inversions of iterated random-to-top shuffles in two asymptotic regimes. Our proofs are analytic, and they utilize new combinatorial decompositions that represent each…

We prove that almost all solutions of the Markoff-Hurwitz equation over a residue field modulo $p$ can be obtained from one another by a chain of natural transformations. We also study recurrence sequences considered modulo prime $p$.

In this paper, we consider the moments of statistics on conjugacy classes of the colored permutation groups $\mathfrak{S}_{n,r}=\mathbb{Z}_r\wr \mathfrak{S}_n$. We first show that any fixed moment coincides on all conjugacy classes where…