## The Number of Self-Conjugate Irreps On a Finite Group of Odd Order

**Point of post: **In this post we use our recent works on the number of self-conjugate irrep classes of a finite group to show that for finite groups of odd order every non-trivial irrep is complex.

*Motivation*

We’ve done much recent work on finding different characterizations of the number of self-conjugate irrep classes of a finite group. In particular, we’ve show that if is a finite group and denotes the number of self-conjugate irreps classes of then

where is the square root function for . Thus, if one knew entirely the nature of the square root function on then one would know the nature of the number of self-conjugate irrep classes of . Unfortunately, it is hard to say anything about the square root function on an entirely general group–in particular there is no way to calculate the square root function if is a finite group of even order. That said, as we shall see the square root function has a particularly simple description on finite groups of odd order. Pursuant to this simple nature we shall prove that if is a finite group of odd order then the only self-conjugate irrep class of is the trivial irrep class .

*The Behavior of The Square Root Function on Finite Groups off Odd Order*

We begin by showing that the sets in the definition of the square root function, namely the sets form a partition of . Giving us the fundamental relation

Indeed:

**Theorem: ***Let be a group (not necessarily finite) and define, for every , to be the set . Then, forms a partition of .*

**Proof: **Clearly it’s so that forms a cover of since for every one clearly has that . Thus, it remains to prove that for . But, this is relatively clear, for if then one must have that . From this the conclusion follows.

**Corollary:** *Let be a finite group. Then,*

* *

* *

Next, we show that for finite groups of odd order the square root function has a particularly simple behavior, namely:

**Theorem: ***Let be a finite group of odd order, say . Then, for every .*

**Proof: **First note, using the very basic corollary of Lagrange’s theorem we know that for every and so in particular and thus so that for every , in particular for every . But, by our previous theorem we have that

From where it follows that for every . The conclusion follows.

From this we may ascertain our main theorem. Namely:

**Theorem: ***Let be a finite group of odd order, then every non-trivial irrep of is complex.*

**Proof: **From our previous theorem we have that for every . Thus, by our recent theorem if denotes the number of self-conjugate irrep classes of , then

But, since the trivial class containing the trivial irrep is always self-conjugate we may conclude that for every other one has that is not self-conjugate, and thus complex. The conclusion follows.

**References:**

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

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

[…] it is to prove using representation theory. Namely, it almost falls out from our previous theorem that every non-trivial irrep of a group of finite order is […]

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