Category-Theoretic relation between Orbit-Stabilizer and Rank-Nullity Theorems In linear algebra, the Rank-Nullity theorem states that given a vector space $V$ and an $n\times n$ matrix $A$,
$$\text{rank}(A) + \text{null}(A) = n$$
or that
$$\text{dim(image}(A)) + \text{dim(ker}(A)) = \text{dim}(V).$$

In abstract algebra, the Orbit-Stabilizer theorem states that given a group $G$ of order $n$, and an element $x$ of the set $G$ acts on,
$$|\text{orb}(x)||\text{stab}(x)| = |G|.$$

Other than the visual similarity of the expressions, is there some deeper, perhaps category-theoretic connection between these two theorems? Is there, perhaps, a functor from the category of groups $\text{Grp}$ to some category where linear transformations are morphisms? Am I even using the words functor and morphism correctly in this context?

As was pointed out in the comments by Clement Guerin and Berci above, the Rank-Nullity Theorem is more properly seen as an immediate consequence of the First Isomorphism Theorem, which says that $\mathrm{Im}(A) \cong V / \mathrm{Ker}(A)$. Taking dimensions of these spaces gives the statement of the Rank-Nullity Theorem, since the "rank" is the dimension of the image of $A$, and the "nullity" is the dimension of the kernel, and the dimension of $V / \mathrm{Ker}(A)$ is just the difference in dimensions $\dim(V) - \dim(\mathrm{Ker}(A))$.
