Given a finitely generated $G$-module $M$, and distinct $G$-submodules $A, B, C$ with $C \subset B$. How could we construct a $G$-submodule $D$ explicitly such that $(C+D)/C \cong (A+B)/B$? Does $D$ always exist?
2026-04-01 22:43:08.1775083388
Module construction
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We may replace $M$ by $A+B$, and by quotienting out by $C$, we may assume $C=0$. The question then becomes: given a submodule $B\leq M$, is the map $M\to M/B$ split? The answer in general is 'no'.
Over a field of characteristic not dividing the order of the group (eg over the complex numbers), however, the answer is 'yes' by Maschke's Theorem, which tells us that every $G$-module is semisimple.