On linear version of an elementary group theory result

TL;DR

This paper extends a classical cyclic group theory result to linear algebra, relating primitive subspaces in field extensions and exploring partitions of finite fields.

Contribution

It introduces a linear algebraic analogue of a group theory theorem for primitive subspaces in field extensions, connecting group properties with linear algebra.

Findings

Established a linear version of the group theory relation for primitive subspaces.

Analyzed partitions of finite fields using primitive subspaces.

Connected group-theoretic concepts with linear algebra and field theory.

Abstract

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.