Related papers: When Any Group of N Elements is Cyclic?

The spectrum of a group is the set of its element orders. A finite group $G$ is said to be recognizable by spectrum if every finite group that has the same spectrum as $G$ is isomorphic to $G$. We prove that the simple alternating groups…

In this note, we describe a construction that leads to families of graphs whose critical groups are cyclic. For some of these families we are able to give a formula for the number of spanning trees of the graph, which then determines the…

Denote the sum of element orders in a finite group $G$ by $\psi(G)$ and let $C_n$ denote the cyclic group of order $n$. In this paper, we prove that if $|G|=n$ and $\psi(G)>\frac{13}{21}\,\psi(C_n)$, then $G$ is nilpotent. Moreover, we have…

We prove several results detecting ciclicity or nilpotency of a finite group $G$ in terms of inequalities involving the orders of the elements of $G$ and the orders of the elements of the cyclic group of order $|G|$. We prove that, among…

We present a survey of exact and asymptotic formulas on the number of cyclic subgroups and total number of subgroups of the groups ${\Bbb Z}_{n_1} \times \cdots \times {\Bbb Z}_{n_k}$, where $k\ge 2$ and $n_1,\ldots,n_k$ are arbitrary…

Let $H^\times$ be the group of units of a multiplicatively written monoid $H$. We say $H$ is acyclic if $xyz \ne y$ for all $x, y, z \in H$ with $x \notin H^\times$ or $z \notin H^\times$; unit-cancellative if $yx \ne x \ne xy$ for all $x,…

Let $n>0$ be an integer and $\mathcal{X}$ be a class of groups. We say that a group $G$ satisfies the condition $(\mathcal{X},n)$ whenever in every subset with $n+1$ elements of $G$ there exist distinct elements $x,y$ such that $<x,y>$ is…

In this note we give a new proof of the fact that an elementary subgroup (in the sense of first-order theory) of a non abelian free group $\mathbb{F}$ must be a free factor. The proof is based on definability of orbits of elements of under…

The prime coprime graph $\Theta(G)$ of a finite group $G$ is the graph whose vertex set is $G$ and any two distinct vertices are adjacent if the greatest common divisor of their orders is either $1$ or a prime. In this paper, we investigate…

Let $G$ be a finite group and $\Z G$ its integral group ring. We show that if $\alpha$ is a non-trivial bicyclic unit of $\Z G$, then there are bicyclic units $\beta$ and $\gamma$ of different types, such that $\GEN{\alpha,\beta}$ and…

In this paper we consider two functions related to the arithmetic and geometric means of element orders of a finite group, showing that certain lower bounds on such functions strongly affect the group structure. In particular, for every…

In this article we give an order-dividing bijective function between cyclic and non cyclic groups of finite order. In particular, we prove that there exists a bijective function from D_{2n} to Z_{2n} for any natural integer n; and from Z_p…

We refer to the set of the orders of elements of a finite group as its spectrum and say that groups are isospectral if their spectra coincide. We prove that with the only specific exception the solvable radical of a nonsolvable finite group…

A group K is said to be a B-group if every permutation group containing K as a regular subgroup is either imprimitive or 2-transitive. In the second edition of his influential textbook on finite groups, Burnside published a proof that…

The fundamental group of the complement of a plane curve is a very important topological invariant. In particular, it is interesting to find out whether this group is determined by the combinatorics of the curve or not, and whether it is a…

Say that a finite group $G$ is mixable if a product of random elements, each chosen independently from two options, can distribute uniformly on $G$. We present conditions and obstructions to mixability. We show that $2$-groups, the…

For a finite group $G$, the vertices of the prime graph $\Gamma(G)$ are the primes that divide $|G|$, and two vertices $p$ and $q$ are connected by an edge if and only if there is an element of order $pq$ in $G$. Prime graphs of solvable…

Let $(G,+)$ be an abelian group and consider a subset $A \subseteq G$ with $|A|=k$. Given an ordering $(a_1, \ldots, a_k)$ of the elements of $A$, define its {\em partial sums} by $s_0 = 0$ and $s_j = \sum_{i=1}^j a_i$ for $1 \leq j \leq…

Let $G$ be a finite group, $L_1(G)$ be its poset of cyclic subgroups and consider the quantity $\alpha(G)=\frac{|L_1(G)|}{|G|}$. The aim of this paper is to study the class $\cal{C}$ of finite nilpotent groups having…

This paper addresses the problem of finding $Q_{m,t}\left(n\right)$, the number of possible ways to partition any member $n$ of the cyclic group $\mathbb{Z}/m\mathbb{Z}$ into $t$ distinct parts. When $m$ is odd, it was previously known that…