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 n+1-cover if and only if n is even and G (C2)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 n10 we find all pairs (m,p) (m0 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).