Minimal blocking sets in PG(n,2) and covering groups by subgroups

Details Minimal blocking sets in PG(n,2) and covering groups by subgroups Minimal blocking sets in PG(n,2) and covering groups by subgroups

2007-08-16

arxiv.org/abs/0708.2282v1

Mathematics

Group Theory

Scientific paper

In this paper we prove that a set of points $B$ of PG(n,2) is a minimal blocking set if and only if $=PG(d,2)$ 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 $=PG(m,p)$.