Relation between finite Abelian Groups and traces?

Question

I have recently read Kronecker's 1870 paper on finite Abelian groups, on the definition of abstract group and so on. It turns out that such definition is literally taken over (being probably unaware of it, for I am almost certain they did not consult the original source, but rather learnt about it from Noether, Van der Waarle or the like) by a couple of very prominent linguists which then proceed, within their theoretical framework, to define traces of operators. I would like to know in which specific way (if any) abstract groups and traces (as in traces of a tensor or a matrix) are connected mathematically. Thanks in advance.

Answer 1

Cameron Williams is referring to character theory. I'll preface all of this by saying that irreducible characters of a finite abelian group $A$ are merely group homomorphisms $\lambda : A \to (\Bbb F, \times)$ from $A$ to the multiplicative group of a field $\Bbb F$, the usual field being $\Bbb C$. But in general, you need much heavier machinery.

Essentially, given a(n abstract or otherwise) group $G$, there are a number of ways to study $G$. Often times it's most illuminating to consider the relationships $G$ can have with other groups, and usually this involves studying homomorphisms $\phi : G \to H$; maps to another group $H$ that 'preserve the group structure' so that $\phi(gg') = \phi(g)\phi(g')$.

Of course there may be lots of groups $H$ to choose from! A particularly fruitful choice is the general linear group $GL(V)$ of invertible linear transformations $V \to V$ of a finite-dimensional vector space $V$. If the vector space $V$ is over a field $\Bbb F$, we call such a homomorphism

$$\mathcal{X}: G \to GL(V)$$

an $\Bbb F$-representation of $G$. Classically our vector space is over the complex numbers, so that $V = \Bbb C^n$ for some integer $n$.

This works out fantastically well, because vector spaces and linear transformations are very well understood, compared to an arbitrary abstract group.

But it turns out that the full representation could be condensed down: Given an $\Bbb F$-representation $\mathcal{X}: G \to GL(V)$, we define the $\Bbb F$-character afforded by $\mathcal{X}$ by

\begin{align*} \chi : G &\to \Bbb F\\ g &\mapsto tr\left(\mathcal{X}(g)\right). \end{align*}

Thus, instead of needing the representation itself, we need only the traces of the images of group elements under a representation. The set of irreducible characters of a group $G$ can reveal fantastic amounts of information about a group, but at a reduced cost compared to representations.

In some sense, character theory tells us even more about groups than group theory does! For example, Burnside's Theorem, a purely group-theoretic result that groups of order $p^aq^b$ are solvable with $p, q$ prime and $a, b$ positive integers, went over 50 years with no known proofs that didn't use group characters. This question on MO points to a result about the Frobenius kernel that has gone over 100 years without a purely group-theoretic proof; every known proof uses character theory.