I am now going to look at some consequences of the theorem we proved in part (1).

1) Let be a finite abelian group. Then, by the corollary in this post, the number of irreducible representations of is and all such representations have degree one by the theorem in that post. The theorem in part (1) gives the same result: since is abelian, the number of conjugacy classes of is Also, since is abelian, in the identity in the proof of that theorem, we must have

2) A finite group is abelian if and only if every irreducible representation of has degree one.

*Proof.* Well, is abelian if and only if is abelian. Now in the proof of the theorem in part (1) shows that is abelian if and only if

3) The number of (pairwise non-equivalent) irreducible representations of is One of these representations has degree two and the rest have degree one.

*Proof*. By the theorem in part (1), we have We also saw in the second remark in here that the number of degree one representations of is two. So and hence the only possibility now is that and This also proves the algebra isomorphism

4) The number of (pairwise non-equivalent) irreducible representations of quaternion group is One of these representations has degree two and the rest have degree one.

*Proof*. Again by the theorem in part (1), we have We also proved in the third remark in here that has exactly four representations of degree one. So Clearly, the only possibility now is that and This also proves the algebra isomorphism

5) The number of (pairwise non-equivalent) irreducible representations of the dihedral group of order eight is One of these representations has degree two and the rest have degree one.

*Proof*. See the fourth remark in here and copy the above proof for We also get

**Remark**. By 4) and 5), although See also here.