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?

The orbit-stabilizer formula stems from the fact that, given an action
of $G$ on $X$, the map
$$ G/G_x \ni gG_x \mapsto gx \in Gx $$
is an isomorphism of $G$-sets between the coset space $G/G_x$ and
the orbit $Gx$ of $x \in X$. Here $G$ acts on $G/G_x$ via left
multiplication.
The closest analogy is the case of an $R$-module $M$ where
$R/ann(m) \cong Rm$ for any $m \in M$.
