$kG$-modules $0 \to M_0 \to M_1 \to M_2 \to 0$ we get $$\dim^{{\operatorname{Elek}}}_{kG}(M_1) = \dim^{{\operatorname{Elek}}}_{kG}(M_0) + \dim^{{\operatorname{Elek}}}_{kG}(M_2);$$

3.  If the finitely generated $kG$-module $M$ satisfies $\dim^{{\operatorname{Elek}}}_{kG}(M) = 0$, then every quotient module $Q$ of $M$ satisfies $\dim^{{\operatorname{Elek}}}_{kG}(Q) = 0$.

The first two assertions are proved in [@Elek(2003c) Theorem 1]. Notice that the third condition does not necessarily follow from additivity since the kernel of the epimorphism $M \to Q$ may not be finitely generated. But the third statement is a direct consequence of the definition of Elek’s dimension.

\[rem:Pontrjagin\_duality\] Identify the left $kG$-module $kG^n$ with the finitely supported functions in ${\operatorname{map}}(G,k^n)$. Here we view ${\operatorname{map}}(G,k^n)$ as a left $k$-vector space by $(af)(g) = af(g)$, and the left $G$-action is given by $(hf)(g) = f(h^{-1}g)$ for $h,g \in G$ and $a \in k$. Let $$    \langle\_,\_\rangle\colon kG^n \times {\operatorname{map}}(G, k^n)$$ be the canonical pairing (evaluation) of $kG^n$ and its dual ${\operatorname{map}}(G, k^n)$. If we view an element $f\in kG^n$ as a finitely supported function $G\to k^n$ (in ${\operatorname{map}}(G,k^n)$), then the pairing of $f\in kG^n$ with $l\in {\operatorname{map}}(G, k^n)$ is given by $$        \langle f,l\rangle=\sum_{g\in G}(f(g),l(g)),$$ where $(\_,\_)$ denotes the standard inner product in $k^n$. For a subset $W\subset kG^n$ let $$W^\perp 