  In this paper 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 $\mathfrak{C}_{n+1}$-cover if and only
if $n$ is even and $G\cong (C_2)^{n}$, where by a $\mathfrak{C}_m$-cover for a
group $H$ we mean a set $\mathcal{C}$ of size $m$ of maximal subgroups of $H$
whose set-theoretic union is the whole $H$ and no proper subset of
$\mathcal{C}$ 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 $<B>=PG(m,p)$.
