I have a question about the Maschke's theorem in the group representation.
I know that Maschke's theorem says that "Every representation of a finite group having positive degree is completely reducible, i.e. it has a direct sum of irreducible G-modules.".
I'm very confused when a G-module is different to the given G-module.
More precisely, if G is a finite group and V is a G-module with $\dim V>0$, then V is a direct sum of irreducible G-submodules $V_i$. Now, if W is another G-module, then by the Maschke's theorem, W also has a decomposition of irreducible G-submodules. I wonder if the irreducible G-submodules of W are precisely $V_i$. That is, I think the irreducible factors does not depend on G-module, and depend only on the given group G.
Is it right? I need your help. Thanks.