Analogy between Lagrange's Theorem and Rank-Nullity Theorem?

Question

One can view view Lagrange's Theorem $$|G/H|=|G|/|H|$$ and the Rank-Nullity Theorem $$\dim(V/U)=\dim(V)-\dim(U)$$ as directly analogous. Does anyone know a high-level explanation of this analogy? I suspect it may involve the word "valuation" or the word "Artinian".

Answer 1

As Igor Rivin suggests, the explanation involves the word "logarithm". In fact, if $\mathfrak C$ is the category of groups, you can define an invariant of the category \begin{align*} \log\lvert{-}\rvert:Ob(\mathfrak C)&\to \mathbb R_{\geq 0}\cup\{\infty\}\\ M&\mapsto \log \lvert M\rvert \end{align*} This invariant is additive, in the sense that, given a surjective morphism $\phi:M\to N$ in the category, $\log\lvert M\rvert=\log\lvert\ker(\phi)\rvert+\log\lvert N\rvert$.

In any category you can give analogously the definition of additive invariant. If your category is Abelian the usual thing is to say that an invariant $i:Ob(\mathfrak C)\to \mathbb R_{\geq 0}\cup\{\infty\}$ is additive provided $i(B)=i(A)+i(C)$ for any short exact sequence $0\to A\to B\to C\to 0$.

The dimension of vector spaces and the composition length of modules are examples of such additive invariants.

There is another property that the above invariants satisfy, that is, they commute with direct unions, that is, if $$M=\underset{\to}{\bigcup}M_\alpha$$ then, $i(M)=\sup_\alpha i(M_\alpha)$. Such an invariant is called upper continuous. In general, additive and upper continuous invariants are called length functions (as they generalize the usual composition length).

There are some attempts in the literature to classify in some sense all the additive invariants of some category, typically a category of modules. See for example:

Northcott, D. G.; Reufel, M. A generalization of the concept of length. Quart. J. Math. Oxford Ser. (2) 16 (1965) 297-–321. MR195905

Vámos, P. Additive functions and duality over Noetherian rings. Quart. J. Math. Oxford Ser. (2) 19 (1968) 43–-55. MR223434