On linear version of an elementary group theory result

Abstract

Given a cyclic group G of order p r , where p is a prime and r ∈ N. It is well-known that the order of its greatest proper subgroup ψ(G) and the number of its generating elements φ(G) satisfy ψ(G) + φ(G) = p r . In this paper, we study a linear version of this group theory theorem for primitive subspaces in a field extension using some linear algebra tools. We also briefly discuss partitions of finite fields by using their primitive subspaces.Notation: The cardinality of a set S shall be denoted by #S. For a vector space V over a field K, we denote its dimension by dimK V . The number of positive divisors of a positive integer n is denoted by d(n). For simplicity of notation, A ⊂ B means that A is a proper subset of B.