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Let $\mathbb{F}_q$ be the field with $q$ elements and of characteristic $p$.For $a\in\mathbb{F}_p$ consider the set \begin{equation*}  S_a(n)=\{f\in\mathbb{F}_q[x]\mid\operatorname{deg}(f)=n,~f\text{ irreducible,monic and} \operatorname{Tr}(f)=a\}. \end{equation*} In a recent paper, RobertGranger proved for $q=2$ and $n\ge 2$ that $|S_1(n)|-|S_0(n)|= 0$ if $2\nmid n$and $|S_1(n)|-|S_0(n)|=|S_1(n/2)|$ if $2\mid n$.  We will prove a generalization of this result for all finite fields. This ispossible due to an observation about the size of certain subsets of monicirreducible polynomials arising in the context of a group action of subgroupsof $\operatorname{PGL}_2(\mathbb{F}_q)$ on monic polynomials. Additionally, itenables us to apply these methods to prove two further results that are verysimilar in nature.

On the Recursive Behaviour of the Number of Irreducible Polynomials with Certain Properties over Finite Fields