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Algebra Discrete Math., 2010, Volume 9, Issue 2, Pages 108–114 (Mi adm32)  

This article is cited in 3 scientific papers (total in 3 papers)

RESEARCH ARTICLE

Thin systems of generators of groups

Ievgen Lutsenko

Dept. Cybernetics, Kyiv University, Volodymyrska 64, 01033 Kyiv, Ukraine

Abstract: A subset $T$ of a group $G$ with the identity $e$ is called $k$-thin $(k\in{\mathbb N})$ if $|A\cap gA|\leqslant k$, $|A\cap Ag|\leqslant k$ for every $g\in G$, $g\ne e$. We show that every infinite group $G$ can be generated by some 2-thin subset. Moreover, if $G$ is either Abelian or a torsion group without elements of order 2, then there exists a 1-thin system of generators of $G$. For every infinite group $G$, there exist a 2-thin subset $X$ such that $G=XX^{-1}\cup X^{-1}X$, and a 4-thin subset $Y$ such that $G=YY^{-1}$.

Keywords: small, $P$-small, $k$-thin subsets of groups.

Full text: PDF file (211 kB)

Bibliographic databases:
MSC: 20F05, 20F99
Received: 10.03.2010
Revised: 10.03.2010
Language:

Citation: Ievgen Lutsenko, “Thin systems of generators of groups”, Algebra Discrete Math., 9:2 (2010), 108–114

Citation in format AMSBIB
\Bibitem{Lut10}
\by Ievgen Lutsenko
\paper Thin systems of generators of groups
\jour Algebra Discrete Math.
\yr 2010
\vol 9
\issue 2
\pages 108--114
\mathnet{http://mi.mathnet.ru/adm32}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2808784}
\zmath{https://zbmath.org/?q=an:1209.20033}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Petrenko O., Protasov I.V., “Thin Ultrafilters”, Notre Dame J. Form. Log., 53:1 (2012), 79–88  crossref  mathscinet  zmath  isi  elib  scopus
    2. Protasov I.V., “Thin Subsets of Topological Groups”, Topology Appl., 160:10 (2013), 1083–1087  crossref  mathscinet  zmath  isi  elib  scopus
    3. Banakh T.O., Protasov I.V., Slobodianiuk S.V., “Scattered Subsets of Groups”, Ukr. Math. J., 67:3 (2015), 347–356  crossref  mathscinet  zmath  isi  elib  scopus
  • Algebra and Discrete Mathematics
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