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I'm reading about the history of character theory of finite group, especially about the invention of character theory by Frobenius.

According to most of the related papers (e.g. Pioneers of Representation Theory by C.W.Curtis), the character of abelian groups occurs earlier, which was defined as a homomorphism from an abelian group $G$ to $C^{\times}$. And it is the problem of factorization of group determinants which motivates Frobenius' generalization of characters to arbitrary finite group, as is described in the introduction of one of the Frobenius' papers Über Gruppencharaktere.

Here is Frobenius' original definition: Let $G$ be a finite group. For each conjugacy class $C_{i}$ of $G$, let $h_{ijk}$ be the number of solutions of the equation of the equations $abc = 1$ for $a\in C_{i}, b\in C_{j}, c\in C_{k}$. Denote as $C_{i^{'}}$ the conjugacy class consisting of the inverses of elements in $C_{i}$, and put $$a_{ijk}=\frac{h_{i^{'}jk}}{|C_{i}|}.$$ Via solving the 'structure equations' $$r_{j}r_{k}=a_{ijk}r_{i},$$ Frobenius got $n$ different solutions $(r_{q1}, r_{q2}, ..., r_{qn})$. Then he defined a character of a finite group $G$ as a class function (constant on every conjugacy class $C_{i}$ ) $\chi$ , taking for any $g\in C_{i}$ the value $$\chi_{q}(g) = \frac{fr_{qi}}{|C_{i}|},$$ where ${r_{qi}}$ is given above by the structure equations and $f$ is used to normalize $\chi_{q}$ in order that it would satisfy the orthogonality relations.

This definition seems to be understandable in modern context of representation theory. What seems highly unnatural to me is that this definition is triggered by the attempt to factorize the group determinant.

First of all, the coefficients of linear factors of the group determinant are exactly the linear characters of the group, and any other factor of the determinant has a degree more than $1$ and can't be factored into linear forms, so there seems to be no reason to generalize characters to the nonabelian case given that the linear factors are the only to appear in the factorization.

Furthermore, I can't see how intuitively this problem of factorization could provide any clue leading to an explicit definition of group character like above. It seems that Frobenius' idea was to construct a homomorphism to $C^{\times}$ from an algebra whose elements are the conjugacy classes with the operation induced by usual multiplication law of a group. I do understand what this algebra is in the modern context, but I can't understand how the latter was motivated by the factorization problem.

Last but not least, Frobenius chose to base his definition on 'hypercomplexes' (seems equivalent to the modern concept of an algebra). I can't see the relation between the 'hypercomplexes' and the factorization problem.

Will anyone be so kind to explain the intuition behind this definition, given the context of group determinants? Thanks in advance!

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We can’t redo the history. It just happens to be the case that Dedekind’s questions to Frobenius about group determinants were the original inspiration. It is not intuitive. Only later did Frobenius find a more conceptual approach, the one we use today. Group determinants were abandoned since they are not why we care about group characters and they are a strange motivation, as you have already seen. Maybe the article https://kconrad.math.uconn.edu/articles/groupdet.pdf will be helpful.

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  • $\begingroup$ Thank you very much. I can only express my appreciation by adding a comment because I'm new to this community and have no permission to vote for you. $\endgroup$
    – zyy
    Apr 11 at 13:35

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