# Abstract Nonsense

## 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 $G$ is a finite group and $\mathfrak{s}$ denotes the number of self-conjugate irreps classes of $G$ then

$\displaystyle \mathfrak{s}=\frac{1}{|G|}\sum_{g\in G}\sqrt{g}\;^2$

where $\sqrt{\text{ }}$ is the square root function for $G$. Thus, if one knew entirely the nature of the square root function on $G$ then one would know the nature of the number of self-conjugate irrep classes of $G$. 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 $G$ 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 $G$ is a finite group of odd order then the only self-conjugate irrep class of $G$ is the trivial irrep class $\alpha_\text{triv}$.