Bases in finite groups of small order
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]>\sq...
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| Format: | Article |
| Language: | English |
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Vasyl Stefanyk Precarpathian National University
2021-06-01
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| Series: | Karpatsʹkì Matematičnì Publìkacìï |
| Subjects: | |
| Online Access: | https://journals.pnu.edu.ua/index.php/cmp/article/view/4887 |
| _version_ | 1797205818129514496 |
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| author | T.O. Banakh V.M. Gavrylkiv |
| author_facet | T.O. Banakh V.M. Gavrylkiv |
| author_sort | T.O. Banakh |
| collection | DOAJ |
| description | 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 $\le 40$. |
| first_indexed | 2024-04-24T08:57:10Z |
| format | Article |
| id | doaj.art-0f515499976b4bb48ebc9c8c9d640a96 |
| institution | Directory Open Access Journal |
| issn | 2075-9827 2313-0210 |
| language | English |
| last_indexed | 2024-04-24T08:57:10Z |
| publishDate | 2021-06-01 |
| publisher | Vasyl Stefanyk Precarpathian National University |
| record_format | Article |
| series | Karpatsʹkì Matematičnì Publìkacìï |
| spelling | doaj.art-0f515499976b4bb48ebc9c8c9d640a962024-04-16T07:05:54ZengVasyl Stefanyk Precarpathian National UniversityKarpatsʹkì Matematičnì Publìkacìï2075-98272313-02102021-06-0113114915910.15330/cmp.13.1.149-1594254Bases in finite groups of small orderT.O. Banakh0https://orcid.org/0000-0001-6710-4611V.M. Gavrylkiv1https://orcid.org/0000-0002-6256-3672Ivan Franko Lviv National University, 1 Universytetska str., 79000, Lviv, Ukraine; Institute of Mathematics, Jan Kochanowski University in Kielce, 7 Uniwersytecka str., 25406, Kielce, PolandVasyl Stefanyk Precarpathian National University, 57 Shevchenka str., 76018, Ivano-Frankivsk, UkraineA 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 $\le 40$.https://journals.pnu.edu.ua/index.php/cmp/article/view/4887finite groupabelian groupbasisbasis sizebasis characteristic |
| spellingShingle | T.O. Banakh V.M. Gavrylkiv Bases in finite groups of small order Karpatsʹkì Matematičnì Publìkacìï finite group abelian group basis basis size basis characteristic |
| title | Bases in finite groups of small order |
| title_full | Bases in finite groups of small order |
| title_fullStr | Bases in finite groups of small order |
| title_full_unstemmed | Bases in finite groups of small order |
| title_short | Bases in finite groups of small order |
| title_sort | bases in finite groups of small order |
| topic | finite group abelian group basis basis size basis characteristic |
| url | https://journals.pnu.edu.ua/index.php/cmp/article/view/4887 |
| work_keys_str_mv | AT tobanakh basesinfinitegroupsofsmallorder AT vmgavrylkiv basesinfinitegroupsofsmallorder |