Related papers: Adding generators in cyclic groups

The power graph of a group $G$ is a graph with vertex set $G$, where two distinct vertices $a$ and $b$ are adjacent if one of $a$ and $b$ is a power of the other. Similarly, the enhanced power graph of $G$ is a graph with vertex set $G$,…

In this paper, we introduce a novel approach for generating random elements of a finite group given a set of generators of that. Our method draws upon combinatorial group theory and automata theory to achieve this objective. Furthermore, we…

Let $n$ be a large number. A subset $A$ of $Z_n$ is complete if $S_A = Z_n$, where $S_A$ is the collection of the subset sums of $A$. Olson proved that if $n$ is a prime and $|A|> 2n^{1/2} $, then $S_A$ is complete. We show that a similar…

We examine a class of geometric theorems on cyclic 2n-gons. We prove that if we take n disjoint pairs of sides, each pair separated by an even number of polygon sides, then there is a linear combination of the angles between those sides…

Let G be a unipotent algebraic subgroup of some GL_m(C) defined over Q. We describe an algorithm for finding a finite set of generators of the subgroup G(Z) = G \cap GL_m(Z). This is based on a new proof of the result (in more general form…

Let $G$ be a finite group and $\psi(G) = \sum_{g \in G} o(g)$, where $o(g)$ denotes the order of $g \in G$. In [M. Herzog, et. al., Two new criteria for solvability of finite groups, J. Algebra, 2018], the authors put forward the following…

The set of all centralizers of elements in a finite group $G$ is denoted by $Cent(G)$ and $G$ is called $n-$centralizer if $|Cent(G)| = n$. In this paper, the structure of centralizers in a non-abelian finite group $G$ with this property…

We show that for a subset $A$ of the cyclic group of prime order $p>3$, if the sumset $A+A-2A$ is not the whole group, then $|A|\le \frac27\,p$. Besides combinatorial arguments, we utilize a general technique involving linear programming.

We present a sharp upper bound for the number of generators of a finite group in terms of the ratio between the order and the exponent.

Let $\mathcal{A}$ be the group algebra $\mathbf{k}[S_n]$ of the $n$-th symmetric group $S_n$ over a commutative ring $\mathbf{k}$. For any two subsets $A$ and $B$ of $[n]$, we define the elements \[ \nabla_{B,A}:=\sum_{\substack{w\in…

Let A be an infinite set of generators for a group G, and let L_A(r) denote the number of elements of G whose word length with respect to A is exactly r. The purpose of this note is to determine all growth functions L_A(r) associated to…

We describe an algorithm for obtaining generators of the unit group of the integral group ring ZG of a finite abelian group G. We used our implementation in Magma of this algorithm to compute the unit groups of ZG for G of order up to 110.…

In this article, we study connections between components of the Cayley graph $\mathrm{Cay}(G,A)$, where $A$ is an arbitrary subset of a group $G$, and cosets of the subgroup of $G$ generated by $A$. In particular, we show how to construct…

The dominant theme of this thesis is the construction of matrix representations of finite solvable groups using a suitable system of generators. For a finite solvable group $G$ of order $N = p_{1}p_{2}\dots p_{n}$, where $p_{i}$'s are…

Several musical scales, like the major scale, can be described as finite arithmetic sequences modulo octave, i.e. chunks of an arithmetic sequence in a cyclic group. Hence the question of how many different arithmetic sequences in a cyclic…

This paper provides a bridge between two active areas of research, the spectrum (set of element orders) and the power graph of a finite group. The order sequence of a finite group $G$ is the list of orders of elements of the group, arranged…

A group $G$ is said to be $\frac{3}{2}$-generated if every nontrivial element belongs to a generating pair. It is easy to see that if $G$ has this property then every proper quotient of $G$ is cyclic. In this paper we prove that the…

Let $G$ be a finite, non-abelian group of the form $G = A N$, where $A \leq G$ is abelian, and $N \trianglelefteq G$ is cyclic. We prove that the commuting graph $\Gamma(G)$ of $G$ is either a connected graph of diameter at most four, or…

We associate each endomorphism of a finite cyclic group with a digraph and study many properties of this digraph, including its adjacent matrix and automorphism group.

We prove a general divisibility theorem that implies, e.g., that, in any group, the number of generating pairs (as well as triples, etc.) is a multiple of the order of the commutator subgroup. Another corollary says that, in any associative…