相关论文: On invariable generation of alternating groups by …

A group $G$ is invariably generated if there exists a subset $S \subseteq G$ such that, for every choice $g_s \in G$ for $s \in S$, the group $G$ is generated by $\{ s^{g_s} \mid s \in S \}$. In [GGJ16] Gelander, Golan and Juschenko showed…

Two permutations of the natural numbers diverge if the absolute value of the difference of their elements in the same position goes to infinity. We show that there exists an infinite number of pairwise divergent permutations of the…

It is shown that each subgroup of odd index in an alternating group of degree at least 10 has all insoluble composition factors to be alternating. A classification is then given of 2-arc-transitive graphs of odd order admitting an…

It is proved that for any finite dimensional representation of a prime order group over the field of rational numbers, polynomial invariants of degree at most $3$ separate the orbits. A result providing an upper degree bound for separating…

We define a deterministic family of permutations generated by an alternating center-edge extraction process on the ordered set [n] = {1,2,...,n}. Starting from the ordered list (1,2,...,n), one repeatedly removes the median element or…

We prove a result on the asymptotic proportion of randomly chosen pairs of permutations in the symmetric group $S_n$ which "invariably" generate a nonsolvable subgroup, i.e., whose cycle structures cannot possibly both occur in the same…

We derive new formulas for the number of unordered (distinct) factorizations with $k$ parts of a positive integer $n$ as sums over the partitions of $k$ and an auxiliary function, the number of partitions of the prime exponents of $n$,…

In this paper we study left and right 4-Engel elements of a group. In particular, we prove that $<a, a^b>$ is nilpotent of class at most 4, whenever $a$ is any element and $b^{\pm 1}$ are right 4-Engel elements or $a^{\pm 1}$ are left…

A theorem of Weyl tells us that the Lorentz (and parity) invariant polynomials in the momenta of $n$ particles are generated by the dot products. We extend this result to include the action of an arbitrary permutation group $P \subset S_n$…

H\'ethelyi and K\"ulshammer showed that the number of conjugacy classes $k(G)$ of any solvable finite group $G$ whose order is divisible by the square of a prime $p$ is at least $(49p+1)/60$. Here an asymptotic generalization of this result…

A modified Lagrange Polynomial is introduced for polynomial extrapolation, which can be used to estimate the equally spaced values of a polynomial function. As an example of its application, this article presents a prime-generating…

Let $p$ be a prime and $a$ a quadratic non-residue $\bmod p$. Then the set of integral solutions of the diophantine equation $x_0^2 - ax_1^2 -px_2^2 + apx_3^2=1$ form a cocompact discrete subgroup $\Gamma_{p,a}\subset SL(2,\mathbb{R})$ and…

We formulate a relationship between finite-order rondle invariants with respect to triple-point modifications and the lower central series of subgroups of a pure twin group. Using our formulation, we construct infinitely many infinite…

We examine the existence of universal elements in classes of infinite abelian groups. The main method is using group invariants which are defined relative to club guessing sequences. We prove, for example: Theorem: For $n\ge 2$, there is a…

In combinatorics of words, a concatenation of $k$ consecutive equal blocks is called a power of order $k$. In this paper we take a different point of view and define an anti-power of order $k$ as a concatenation of $k$ consecutive pairwise…

Let $K$ be a prime knot in $S^3$ and $G(K)=\pi_1(S^3-K)$ the knot group. We write $K_1 \geq K_2$ if there exists a surjective homomorphism from $G(K_1)$ onto $G(K_2)$. In this paper, we determine this partial order on the set of prime knots…

Given a prime power $p^d$ with $p$ a prime and $d$ a positive integer, we classify the finite groups $G$ with $p^{2d}$ dividing $|G|$ in which all subgroups of order $p^d$ are complemented and the finite groups $G$ having a normal…

In this note, we give an upper bound for the number of elements from the interval $[1,p^{1/4e^{1/2}+\epsilon}]$ necessary to generate the finite field $\mathbb{F}_{p}$ with $p$ an odd prime.

Let p be a prime number. Let G be a finite abelian p-group of exponent n (written additively) and A be a non-empty subset of $]n[:= \{1,2,..., n\}$ such that elements of A are incongruent modulo p and non-zero modulo p. Let $k \geq…

Let A_n be the alternating group on n letters. For n>5, we describe the elements alpha, beta in A_n when alpha^{A_n} beta^{A_n} is the union of at most four distinct conjugacy classes.