On the maximal number of non-overlapping Klein 4-groups inside an elementary abelian 2-group

Abstract

Let $G$ be a finite elementary abelian $2$-group of order $2^n$, for some integer $n \geq 2$. Let $b_n$ be the maximal cardinality of a set ${\mathcal S}$ of subgroups of $G$ such that each member of ${\mathcal S}$ is isomorphic to the Klein $4$-group and any two distinct members of ${\mathcal S}$ meet only in $0$. It is proved that $b_{n+2} \geq 4b_n$. Consequently, $b_n \geq 2^{n-2}$ if $n$ is even, while $b_n \geq 2^{n-3}$ if $n$ is odd; these results are best possible since $b_2=1=b_3$.