Minimal Blocking Sets in PG(n, 2) and Covering Groups by Subgroups |
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Authors: | A Abdollahi M J Ataei A Mohammadi Hassanabadi |
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Institution: | 1. Department of Mathematics , University of Isfahan , Isfahan, Iran a.abdollahi@math.ui.ac.ir;3. Department of Mathematics , University of Isfahan , Isfahan, Iran |
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Abstract: | In this article we prove that a set of points B of PG(n, 2) is a minimal blocking set if and only if ?B? = PG(d, 2) with d odd and B is a set of d + 2 points of PG(d, 2) no d + 1 of them in the same hyperplane. As a corollary to the latter result we show that if G is a finite 2-group and n is a positive integer, then G admits a ? n+1-cover if and only if n is even and G? (C 2) n , where by a ? m -cover for a group H we mean a set 𝒞 of size m of maximal subgroups of H whose set-theoretic union is the whole H and no proper subset of 𝒞 has the latter property and the intersection of the maximal subgroups is core-free. Also for all n < 10 we find all pairs (m,p) (m > 0 an integer and p a prime number) for which there is a blocking set B of size n in PG(m,p) such that ?B? = PG(m,p). |
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Keywords: | Blocking sets Covering groups by subgroups Maximal irredundant covers for groups Projective spaces |
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