We are now going to explain how we can find a representation of from a representation of a subgroup of

**Definition**. Let be a finite group and let be a subgroup of Let be a representation of Let and define by for all and We call the representation **induced** from to and we write

The first thing to do is to see why the definition basically makes sense. I will explain this in the following remarks.

**Remark 1. **Recall** **that if and are rings, is an -bimodule and is a left -module, then is a left -module. The left action of on is defined by for all and Of course, as usual, the action extends to all elements of by linearity. Now, in the above definition, we know that is a left -module (see Remark 1 in here if you have already forgotten!) and obviously is an -bimodule because is a subalgebra of Thus is indeed a -module and the left action of on is defined by for and This is why we defined as you see.

**Remark 2**. It is Clear that Also, we have and for all and Thus and so is a group homomorphism, i.e. is a representation of

In the second part of this note, I will find a formula for in terms of