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

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 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). [ABSTRACT FROM AUTHOR]