Related papers: Finite order elements in the integral symplectic g…

The spectra of a finite group is the set of its element orders. We obtain an arithmetic description of finite symplectic and orthogonal groups. In particular, a description of spectra of all finite simple simplectic and orthogonal groups is…

Let $G$ be a finite $2$-generated non-cyclic group. The spread of $G$ is the largest integer $k$ such that for any nontrivial elements $x_1, \ldots, x_k$, there exists $y \in G$ such that $G = \langle x_i, y\rangle$ for all $i$. The more…

Let $G$ be a finite group of order $n$, and denote by $\rho(G)$ the product of element orders of $G$. The aim of this work is to provide some upper bounds for $\rho(G)$ depending only on $n$ and on its least prime divisor, when $G$ belongs…

Let $G$ be a finite group, and let cs$(G)$ be the set of conjugacy class sizes of $G$. Recalling that an element $g$ of $G$ is called a \emph{vanishing element} if there exists an irreducible character of $G$ taking the value $0$ on $g$, we…

We study the finite solvable groups $G$ in which every real element has prime power order. We divide our examination into two parts: the case $\textbf{O}_2(G)>1$ and the case $\textbf{O}_2(G)=1$. Specifically we proved that if…

A partial order is called semilinear iff the upper bounds of each element are linearly ordered and any two elements have a common upper bound. There exists, up to isomorphism, a unique countable existentially closed semilinear order, which…

Let $\Gamma$ be a finite graph, and for each vertex $i$ let $G_i$ be a finitely presented group. Let $G$ be the graph product of the $G_i$. That is, $G$ is the group obtained from the free product of the $G_i$ by factoring out by the…

Let $o(G)$ be the average order of a finite group $G$. We show that if $o(G)<c$, where $c\in \lbrace \frac{13}{6}, \frac{11}{4}\rbrace$, then $G$ is an elementary abelian 2-group or a solvable group, respectively. Also, we prove that the…

Given an infinite group G, we consider the finitely additive measure defined on finite unions of cosets of finite index subgroups. We show that this shares many properties with the size of subsets of a finite group, for instance we can…

Finite hamiltonian groups are counted. The sequence of numbers of all groups of order $n$ all whose subgroups are normal and the sequence of numbers of all groups of order less or equal to $n$ all whose subgroups are normal are presented.

We prove that the word problem of a finitely generated group $G$ is in NP (solvable in polynomial time by a non-deterministic Turing machine) if and only if this group is a subgroup of a finitely presented group $H$ with polynomial…

Given a finitely generated subgroup $H$ of a free group $F$, we present an algorithm which computes $g_1,\ldots,g_m\in F$, such that the set of elements $g\in F$, for which there exists a non-trivial $H$-equation having $g$ as a solution,…

We are interested in the maximum value achieved by the systole function over all complete finite area hyperbolic surfaces of a given signature $(g,n)$. This maximum is shown to be strictly increasing in terms of the number of cusps for…

In the present paper we study two sequences of real numbers associated to a symplectic diffeomorphism: the uniform norm of the differential of its n-th iteration and the word length of its n-th iteration. In the latter case we assume that…

Let $\rho(n)$ denote the maximal number of different primes that may occur in the order of a finite solvable group $G$, all elements of which have orders divisible by at most $n$ distinct primes. We show that $\rho(n)\leq 5n$ for all $n\geq…

We consider Siegel upper half space of rank two ${\cal H}^2$ and different subgroups $H\subseteq {\bf Sp(4,Z)}$ of finite index. The purpose of this paper is to prove that the field of rational functions of ${\cal H}^2/H$ has general type…

Although $S_\infty$ (the group of all permutations of $\mathbb{N}$) is size continuum, both it and its closed subgroups can be presented as the set of paths through a countable tree. The subgroups of $S_\infty$ that can be presented this…

A group G that is not finitely generated can be written as the union of a chain of proper subgroups. The cofinality spectrum of G, written CF(S), is the set of regular cardinals lambda such that G can be expressed as the union of a chain of…

If V is a representation of a linear algebraic group G, a set S of G-invariant regular functions on V is called separating if the following holds: If two elements v,v' from V can be separated by an invariant function, then there is an f…

The set $\mathcal{R}_{G}(h,k)$ consists of all possible sizes for the $h$-fold sumset of sets containing $k$ elements from an additive abelian group $G$. The exact makeup of this set is still unknown, but there has been progress towards…