TY - JOUR
AU - Banakh, T.O.
AU - Gavrylkiv, V.M.
PY - 2021/06/20
Y2 - 2024/04/19
TI - Bases in finite groups of small order
JF - Carpathian Mathematical Publications
JA - Carpathian Math. Publ.
VL - 13
IS - 1
SE - Scientific articles
DO - 10.15330/cmp.13.1.149-159
UR - https://scijournals.pnu.edu.ua/index.php/cmp/article/view/4887
SP - 149-159
AB - <p>A subset $B$ of a group $G$ is called a <em>basis</em> 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 <em>basis size</em> 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 $\le 40$.</p>
ER -