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 $r[G]>\sqrt{|G|}$. 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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2013 ◽
Vol 88
(3)
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pp. 448-452
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2019 ◽
Vol 19
(10)
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pp. 2050198
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2013 ◽
Vol 56
(3)
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pp. 873-886
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2020 ◽
Vol 0
(0)
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1969 ◽
Vol 10
(3-4)
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pp. 359-362
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