## A ‘Lemma’ (pt. II)

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

*The ‘Lemma’*

We next study as a tool to prove our still looming lemma the multiplicative relation between the elements of the canonical basis of . Namely, we see that:

**Theorem: ***Let be a finite group with conjugacy classes . Then, for any there exists for every such that .*

**Proof: **Recall that is real the center of the group algebra . But, recall that the center of an algebra is a subalgebra. In particular since is a subalgebra of we have that is in particular closed under multiplication–convolution in this case. Thus, since and are class functions then so is . Thus, since is a basis for we know that there does exist constants for such that and in fact (as has already been proven in general) for any . But, by definition one has that

which is surely a nonnegative integer. The conclusion follows.

From this obtain the following:

**Theorem :** *Let be a finite group with conjugacy classes and . Then, if is defined as before then where are, as usual, the algebraic integers.*

**Proof: **We first claim that is a -linear combination of . Indeed, one has that there exists for such that . Thus, applying to both sides and recalling that it’s an algebra homomorphism we may conclude that Fix then . Rephrasing the above,we know that there exists elements of , call them for such that . Let and . A quick check then shows that and so is in the spectrum of some matrix in and so by a prior characterization we know that . Since was arbitrary the conclusion follows.

From this we get the follwing:

**Theorem (The ‘Lemma’):*** Let be a finite group and . Then, for every one has that where is the conjugacy class of containing .*

**Proof: **This follows immediately from the previous theorem and the already proven fact that .

**Corollary: ***Let be a finite group and .Then, either is irrational or .*

So our ‘lemma’ turned out to be a theatrical production in and of itself.

**References:**

1. Isaacs, I. Martin. *Character Theory of Finite Groups*. New York: Academic, 1976. Print**.
**

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