Let $R$ be a commutative ring with $1$. Suppose that I have a short exact sequence $$0\to A\overset{f}{\hookrightarrow}B\overset{g}{\twoheadrightarrow}C\to 0$$ where $f$ is an injection, $g$ is a surjection (of $R-$modules) and $\ker g=\operatorname{Im} f$. Suppose also that each one of these modules has a composition series (wiki).
Can someone tell me how are the three series related? This is a question regarding section 5 of Serre's "Local Fields" book where he explains the Grothendieck groups notion stating the formula $\chi_A(M)=\chi_A(M')\chi_A(M'')$.
The intuitive relation is that the composition series of $A$ and the composition series of $C$ together make a composition series of $B$ (though not necessarily the one you already have).
The composition series of $A$ is part of a composition series of $B$ because $A$ is a submodule, while the composition series of $C$ is part of a composition series of $B$ by the correspondence between submodules of $C\cong B/A$ and submodules of $B$ that contains $A$.
Finally, you connect the two series together by the fact that $\{0\}$ in the composition series of $C$ corresponds in this way exactly to $A$ in the composition series of $A$.