Bases in finite groups of small order

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


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