Minimal Blocking Sets in PG(n, 2) and Covering Groups by Subgroups

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 ℭ _n+1-cover if and only if n is even and G≅ (C ₂) ⁿ , where by a ℭ _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).

Key Words:

• Blocking sets
• Covering groups by subgroups
• Maximal irredundant covers for groups
• Projective spaces

2000 Mathematics Subject Classification:

• 51E21
• 20D60

ACKNOWLEDGMENTS

The authors are grateful to L. Storme for his valuable suggestions on some parts of this article. They are indebted to the referee for pointing out a serious mistake in the original version of the article as well as quoting some useful results from the theory of blocking sets by which the use of GAP has been reduced. This work was supported by the Excellence Center of University of Isfahan for Mathematics.

Notes

Communicated by A. Turull.