Induced representations I am studying representation theory and the induced representations are one thing that I really can't ''grok''.
I was reading Wikipedia article and in the beginning it says: (...) induced representations are rich objects, in the sense that they include or detect many interesting representations.
And it's true that the fact that induced representation of trivial representation is the action on the cosets on the subgroup points in that direction...
I have two questions:


*

*Is there a formalization of this sentence, a theorem of some kind that says which other representations induced representation includes?

*Do you know some good examples/uses of this fact?

 A: The weakest statement one can make is that for any subgroup $H\leq G$ and any complex irreducible representation $\rho$ of $G$, $\rho$ is a summand of some induced representation from $H$. This follows immediately from Frobenius reciprocity.
But there are much stronger results, which go under the bracketing name of "induction theorems". Their common feature is that they say that certain representations of a finite group $G$ are obtained as linear combinations of certain representations induced from certain subgroups.
One of the most important examples is Brauer's induction theorem, which says that all complex characters can be obtained as $\mathbb{Z}$-linear combinations of inductions of 1-dimensional characters from subgroups. Another one is Artin's induction theorem.
Both of these are contained in Serre's book on representation theory, as well as in Isaacs book on character theory. There is a whole chapter on induction theorems in Benson's book Groups and Cohomology, Vol. 1, which also includes some theorems in positive characteristic.
Regarding examples of applications: induction theorems are used a lot in number theory. A prominent example is the application to meromorphicity of Artin L-functions, as explained here, the upshot being that the statement you want to prove behaves well under taking linear combinations of representations and under induction, so you use Brauer's theorem to reduce the statement to one-dimensional characters, when it is much easier to prove. Another such example in number theory is the theory of epsilon factors, as developed by Langlands and Deligne.
