## Irreducible Characters (Pt. II)

**Point of post: **This post is a continuation of this one.

**Irreducible Characters Span Class Functions**

We now show that . Indeed:

**Theorem: ***Let be a finite group then .*

**Proof: **Let . Since in particular and as was already proven. Thus, there exists constants for every and such that

It follows then that for any one has that

which (since the inner most sum is the conjugate of the inner product) is equal to

and since was arbitrary it follows that

And, since was arbitrary the conclusion follows.

From this we get the obvious corollary that where is the number of classes of . From this we can now prove a theorem which, in the future, will help us prove a few interesting structural results about groups. Namely:

**Theorem: ***Let be a finite group. Then, is abelian if and only if every one of it’s irreps has degree one.*

**Proof: **The necessity of this theorem was a previous theorem and so it suffices to show sufficiency. To do this we merely note that if each irrep has degree one then

and thus the number of conjugacy classes of is equal to its order, which is true if and only if is abelian. The conclusion follows.

**References:**

1.Simon, Barry. *Representations of Finite and Compact Groups*. Providence, RI: American Mathematical Society, 1996. Print.

2. Serre, Jean Pierre. *Linear Representations of Finite Groups*. New York: Springer-Verlag, 1977. Print

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