**Lemma**. Let be a finite abelian group. The number of degree one representations of is

*Proof*. Let By the fundamental theorem for finite abelian groups, where each is a cyclic group of order, say, Clearly So every element of is written uniquely as where By Example 1, each has exactly degree one representations. Let be the set of degree one representations of and let be the set of degree one representations of Define by

where If we show that is a bijection, then and the lemma is proved.

1) is well-defined : This is obviouse because if then clearly

2) is injective : If and for all then for any we have

Thus and so is injective.

3) is surjective : Let and define by

for all It is easy to see that Also for all i.e. and thus

**Theorem**. Let be a finite group. The number of degree one representations of is

*Proof .* Since is a finite abelian group, the number of degree one representations of is by the above lemma. So we only need to define a bijection between the set of degree one representations of and the set of degree one representations of Let be the natural homomorphism and define by

It is immediate that is well-defined and injective. So we only need to show that is surjective. To see this, let So is a group homomorphism and hence is abelian because it is isomorphic to a subgroup of Thus

Now define by

for all If then by and hence Thus is well-defined. Clearly is a group homomorphism because is so. Thus Finally for all Hence and the proof is complete.