Self-linked sets in finite groups

Taras Banakh, Volodymyr Gavrylkiv

Published 2017-02-08Version 1

A subset $A$ of a group $G$ is called self-linked if $A\cap gA\ne\emptyset$ for every $g\in G$. The smallest cardinality $|A|$ of a self-linked subset $A\subset G$ is called the self-linked number $sl(G)$ of $G$. In the paper we find lower and upper bounds for the self-linked number $sl(G)$ of a finite group $G$ and prove that $$\frac{1+\sqrt{4|G|+4|G_2|-7}}2\le sl(G)\le \frac{\sqrt[4]{|G|\ln(|G|-1)}+\sqrt{\ln 4}}{\sqrt[4]{|G|\ln(|G|-1)}-\sqrt{\ln 4}}\cdot \sqrt{|G|\ln(|G|-1)}$$where $G_2=\{g\in G:g^{-1}=g\}$ is the set of elements of order at most two in $G$. Also we calculate the self-linked numbers of all Abelian groups of cardinality $\le95$.