On linear version of an elementary group theory result

Given a cyclic group $G$ of order $p^r$, where $p$ is a prime and $r\in\mathbb{N}$. It is well-known that the order of its greatest proper subgroup $\psi(G)$ and the number of its generators $\phi(G)$ satisfy $\psi(G)+\phi(G)=p^r$. In this paper, we give a linear version of this group theory theorem for primitive subspaces in a field extension using tools from field theory and linear algebra. We also discuss partitions of finite fields by using their primitive subspaces.