We really should see an example of inducing a representation. One example we’ll find extremely useful is when we start with the trivial representation.
So, let be a group and be a subgroup. Since this will be coming up a bunch, let’s just start writing for the trivial representation that sends each element of to the matrix . We want to consider the induced representation .
Well, we have a matrix representation, so we look at the induced matrix representation. We have to pick a transversal for the subgroup in . Then we have the induced matrix in block form:
In this case, each “block” is just a number, and it’s either or , depending on whether is in or not. But if , then latex g(t_jH)=(t_iH)$. That is, this is exactly the coset representation of corresponding to . And so all of these coset representations arise as induced representations.