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My question is essentially this: which parts of the representation theory of finite groups are really just applications of module theory, and which are not? Here is an example of each case. Induction of representations (at least for finite groups) is simply extension of scalars for modules over the group ring. Thus the definition of induction and its basic properties can be equally well studied for suitable class of modules with no mention of groups or representations. On the other hand, I have heard Mackey's irreducibility criterion listed as a classic example of a theorem that has no "good" module theoretic generalization because it involves double cosets, and it is very difficult to generalize double coset structure to modules.

Based on these examples I can break my question up into two, more specific questions: 1) What are some other examples of representation theoretic theorems that do or don't generalize well to a module theoretic setting? 2) When, in your opinion, is it useful for a representation theorist to work with the module theory, and when does this merely introduce unnecessary abstraction.

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I would say Clifford's theorem is a group theoretic result since it involved restricting representations to normal subgroups, which doesn't make sense for algebras. A number of theorems about the character table are at least not so natural in the module theory language. The fact that the dimensions of simple modules divide the order of the group and related algebraic integer results are group specific.

Things about faithful representations are group specific (or at least Hopf algebra specific) like that the tensor powers of any faithful rep contains all simples as constituents.

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What if you look at the representation theory of hopf algebras? Then it seems normality makes sense. Also, I don't understand how your last two sentences work, why is this specific to groups? Surely it isn't. – Sean Tilson Nov 28 '11 at 18:18
There are monoids with semisimple algebras such that the dimensions of the simple modules do not divide the order of the monoid. So this is really a group thing. – Benjamin Steinberg Nov 28 '11 at 18:33
I don't know if Clifford theory generalized to Hopf algebras. Good question. – Benjamin Steinberg Nov 28 '11 at 18:42
with respect to your first comment, is it really a group thing or a hopf algebra thing? It seems like a lot of the time an antipode is really what you need rather than just an inverse. – Sean Tilson Nov 28 '11 at 18:54
I am not so familiar with Hopf algebra theory so I am not sure how much generalized to this setting. In any event the OP asked which things about group reps are or are not module theoretic. I think asking which things about group reps are really Hopf algebraic would be a different and interesting question. – Benjamin Steinberg Nov 28 '11 at 19:07

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