Given an elementary abelian $p$-group $G$, it's well known that it can be seen as a vector space over $\mathbb{Z}\_p $. 


But, does someone have an idea about possible sources where I can find proofs that use this fact? 
Examples: linear independence of elements in a group $G $ modulo $\phi(G) $ ($\phi(G)$ is the Frattini group of $G$ ), etc. ...


Does someone have an idea?


Thanks! 



Linear Independence & Group Theory
One example is in the $\sigma$-reduction of a finite group. Denote by $\sigma(|G|)$ the number of prime divisors of $|G|$, and let $\sigma(G)$ denote the maximum number of prime divisors of the order of an element of a group $G$. We call a group $H$ $\sigma$-reduced if every prime $p$ appears in at most one chief factor of $H$. It can be shown that every group $G$ contains a subgroup $H$ so that $\sigma(|G|)=\sigma(|H|)$. (Reference: Huppert's *Character Theory of Finite Groups,* chapter 16. In fact, this chapter also has a few other applications of this concept to Frobenius groups.)


There is an ongoing investigation of a bound on $\sigma(|G|)$ by $\sigma(G)$ for finite solvable groups. (The conjecture is that $\sigma(|G|)\leq 3\sigma(G)$.) After fixing a $\sigma(G)$, it suffices to consider the $\sigma$-reduction of $G$ to obtain the bound on $\sigma(|G|)$ while enjoying the properties of $\sigma$-reduced groups. In particular, Sylow subgroups of the $\sigma$-reduction are elementary abelian, so interplay between the bases of representations over vector spaces of $\mathbb{Z}\_p$ for $p\mid |G|$ play a crucial role. You can see some of this happening in Thomas Keller's earlier papers *Solvable groups with at most four prime divisors in the element orders* and *Solvable groups with a small number of prime divisors in the element orders*.


