Maximal Subgroups Containing given Element

Question

Let $G$ be an elementary abelian $p$-group of finite rank, and $1\neq g\in G$. How do we parametrize the maximal subgroups of $G$, which contain $g$?

Answer 1

Some ideas:

1) Elementary abelian $\,p-$groups are linear spaces over $\,\Bbb F_p:=\Bbb Z/p\Bbb Z\,$ of dimension = the group's rank

2) Maximal subgroups of elem. abelian groups correspond to maximal subspaces of the corresponding linear space.

3) A subspace of a linear space is maximal iff it is the kernel of a non-zero linear functional.

Thus...pant, pant!...You need to find all the non-zero linear functionals s.t. $\,g\,$ belongs to their kernel, i.e if $\,A\,$ is our group, then

$$M\;\text{ is a maximal subgroup of $\,A\,$ containing}\;g\iff \exists\,0\neq\phi\in A^* \;\;s.t.\;\;\phi g=0$$

Where $\,A^*:=\{f:A\to\Bbb F_p\;;\;f\;\text{ is linear}\}\,$