## The Dimension Theorem

**Point of post: **In this post we prove what is called the ‘dimension’ theorem which in essence says that the degree of any irrep of a finite group divides the order of the group.

* *

*Motivation*

So far we’ve obtained some interesting information about the degrees of the irreps of a finite group . We’ve proven that the sum of the degrees squared must equal the order of the group. Also, we’ve proven that the number of degree one irreps of is equal to the order of the abelinization of . In this post we’ll prove the supremely interesting result that the degree of any irrep must divide the order of the group. One of many uses for this will be that we will be able to prove some interesting results about finite groups.

*The Dimension Theorem*

Let as always be a finite group and suppose that we’ve chosen particular representatives and thus irreducible characters have the representation . Our first theorem will show the connection between irreducible characters and algebraic integers. Namely:

**Theorem: ***Let be a finite group then for every and every one has that .*

**Proof: **We merely note that since one has that and so and so by basic matrix analysis every eigenvalue of is a -root of unity and thus trivially an algebraic integer. Thus, being the sum of these algebraic integers (it of course, being the trace of ) is an algebraic integers (since they form a ring).

We now use this to show that for every .

**Theorem: ***Let be a finite group and . Then, .*

**Proof: **Choose an ordering for so that we can list as where . We then define a matrix by . We then define by . We note then by the convolution relations that

and thus is an eigenvalue for . That said, by the previous theorem we know that every entry of is an algebraic integer and thus by previous theorem we may conclude that . But, since evidently we may conclude that and thus as desired.

**References:**

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

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