## Dimension of the Space of Class Functions

**Point of post: **In this post we prove the simple result that the dimension of the space of class functions is equal to the number of conjugacy classes in .

*Motivation*

In our last post we hinted that the dimensionality of the space of class functions of the finite group shall be used to derive a very interestint result. As a step toward this we prove in this post that the dimension of thought of as a subspace of the group algebra is the number of conjugacy classes of .

*Dimension of *

Let be a finite group and the space of all class functions on . Furthermore, let be the distinct conjugacy classes of then:

**Theorem: ***.*

**Proof: **For each let (as usual) denote the indicator function of . We claim that forms a basis for (note that each is evidently a class function itself since it’s constant on the conjugacy classes of ). Clearly is linearly independent since if were such that

then, choosing some representative for we may conclude by the pairwise-disjointness of that

and thus .

Now to see that we merely note that by previous chatracterizations of class functions for each one has that is a singleton say . We claim then that

But, this is indeed obvious since given any we have that for precisely one and

Since was arbitrary 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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