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    Groups containing locally maximal product free sets of size 4

    Anabanti, Chimere (2017) Groups containing locally maximal product free sets of size 4. Working Paper. Birkbeck, University of London, London, UK.

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    Abstract

    Every locally maximal product-free set S in a finite group G satisfies G = S[SS[S−1S[ SS−1 [pS, where SS = {xy| x, y 2 S}, S−1S = {x−1y| x, y 2 S}, SS−1 = {xy−1| x, y 2 S} and pS = {x 2 G| x2 2 S}. To better understand locally maximal product-free sets, Bertram asked whether every locally maximal product-free set S in a finite abelian group satisfy |pS| � 2|S|. This question was recently answered in the negation by the current author. Here, we improve some results on the structures and sizes of finite groups in terms of their locally maximal product-free sets. A consequence of our results is the classification of abelian groups that contain locally maximal product-free sets of size 4, continuing the work of Street, Whitehead, Giudici and Hart on the classification of groups containing locally maximal product-free sets of small sizes. We also obtain partial results on arbitrary groups containing locally maximal product-free sets of size 4, and conclude with a conjecture on the size 4 problem as well as an open problem on the general case.

    Metadata

    Item Type: Monograph (Working Paper)
    Additional Information: Birkbeck Pure Mathematics Preprint Series #38
    Keyword(s) / Subject(s): Product-free sets, locally maximal, maximal, groups
    School: Birkbeck Schools and Departments > School of Business, Economics & Informatics > Economics, Mathematics and Statistics
    Research Centre: Applied Macroeconomics, Birkbeck Centre for
    Depositing User: Administrator
    Date Deposited: 20 Mar 2019 16:50
    Last Modified: 27 Jul 2019 06:02
    URI: http://eprints.bbk.ac.uk/id/eprint/26769

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