Bases in finite groups of small order

Authors

  • T.O. Banakh Ivan Franko Lviv National University, 1 Universytetska str., 79000, Lviv, Ukraine; Institute of Mathematics, Jan Kochanowski University in Kielce, 7 Uniwersytecka str., 25406, Kielce, Poland https://orcid.org/0000-0001-6710-4611
  • V.M. Gavrylkiv Vasyl Stefanyk Precarpathian National University, 57 Shevchenka str., 76018, Ivano-Frankivsk, Ukraine https://orcid.org/0000-0002-6256-3672
https://doi.org/10.15330/cmp.13.1.149-159

Keywords:

finite group, Abelian group, basis, basis size, basis characteristic
Published online: 2021-06-20

Abstract

A subset $B$ of a group $G$ is called a basis of $G$ if each element $g\in G$ can be written as $g=ab$ for some elements $a,b\in B$. The smallest cardinality $|B|$ of a basis $B\subseteq G$ is called the basis size of $G$ and is denoted by $r[G]$. We prove that each finite group $G$ has $r[G]>\sqrt{|G|}$. If $G$ is Abelian, then $r[G]\ge \sqrt{2|G|-|G|/|G_2|}$, where $G_2=\{g\in G:g^{-1} = g\}$. Also we calculate the basis sizes of all Abelian groups of order $\le 60$ and all non-Abelian groups of order $\le 40$.

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How to Cite
(1)
Banakh, T.; Gavrylkiv, V. Bases in Finite Groups of Small Order. Carpathian Math. Publ. 2021, 13, 149-159.