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Mat. Zametki, 2004, Volume 76, Issue 3, Pages 420–426 (Mi mz118)  

This article is cited in 9 scientific papers (total in 9 papers)

Small Systems of Generators of Groups

I. V. Protasov

National Taras Shevchenko University of Kyiv

Abstract: A subset $S$ of a group $G$ is said to be large (left large) if there is a finite subset $K$ such that $G=KS=SK$ $(G=KS)$. A subset $S$ of a group $G$ is said to be small (left small) if the subset $G\setminus KSK$ $(G\setminus KS)$ is large (left large). The following assertions are proved:
  • (1) every infinite group is generated by some small subset;
  • (2) in any infinite group $G$ there is a left small subset $S$ such that $G=SS^{-1}$;
  • (3) any infinite group can be decomposed into countably many left small subsets each generating the group.


DOI: https://doi.org/10.4213/mzm118

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English version:
Mathematical Notes, 2004, 76:3, 389–394

Bibliographic databases:

UDC: 512.536
Received: 21.11.2001
Revised: 01.07.2002

Citation: I. V. Protasov, “Small Systems of Generators of Groups”, Mat. Zametki, 76:3 (2004), 420–426; Math. Notes, 76:3 (2004), 389–394

Citation in format AMSBIB
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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. Lutsenko Ie., Protasov I.V., “Sparse, thin and other subsets of groups”, Internat. J. Algebra Comput., 19:4 (2009), 491–510  crossref  mathscinet  zmath  isi  scopus  scopus
    2. Igor Protasov, “Partitions of groups into sparse subsets”, Algebra Discrete Math., 13:1 (2012), 107–110  mathnet  mathscinet  zmath
    3. Igor Protasov, Sergiy Slobodianiuk, “Prethick subsets in partitions of groups”, Algebra Discrete Math., 14:2 (2012), 267–275  mathnet  mathscinet  zmath
    4. Protasov I.V., “Thin Subsets of Topological Groups”, Topology Appl., 160:10 (2013), 1083–1087  crossref  mathscinet  zmath  isi  scopus  scopus
    5. Banakh T., Protasov I., Slobodianiuk S., “Syndetic Submeasures and Partitions of G-Spaces and Groups”, Int. J. Algebr. Comput., 23:7 (2013), 1611–1623  crossref  mathscinet  zmath  isi  scopus  scopus
    6. Protasov I., Slobodianiuk S., “Partitions of Groups Into Large Subsets”, J. Group Theory, 18:2 (2015), 291–298  crossref  mathscinet  zmath  isi  scopus  scopus
    7. 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  scopus  scopus
    8. Protasov I., Slobodianiuk S., “Partitions of Groups”, Adv. Appl. Discret. Math., 15:1 (2015), 33–60  mathscinet  zmath  isi
    9. Protasov I., Slobodianiuk S., “Note: A conjecture on partitions of groups”, Combinatorica, 36:3 (2016), 365–369  crossref  mathscinet  zmath  isi  elib  scopus
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