If $G$ is a finite group and $\lbrace x\_{g} \rbrace\_{g\in G}$ are commuting formal variables, then one can form a matrix whose $(g,h)$ entry is $x\_{gh^{-1}}$. The determinant of this matrix is a polynomial with integer coefficients and is called the *group determinant*. Considering the factorization of this determinant led Frobenius to discover seminal results in the representation theory of finite groups. [Later on](http://www.jstor.org/stable/2159470), it was shown that a group can be recovered from its group determinant.


Many people have asked me the following question over the years, and I haven't a good answer for it. One might think about looking at something like a generalized determinant over a polynomial ring in infinitely many variables indexed by the group.



> 
> **Question:** In the literature, does there exist a
>  *more or less direct* attempt to generalize the Dedekind-Frobenius
>  group determinant to the setting of
>  infinite groups?
> 
> 
> 


Moreover, if such a thing exists, can it determine the group from which it is constructed?



Is there a Dedekind-Frobenius group determinant for infinite groups?
Such a thing is defined in Steve Humphries' very interesting paper:


Humphries, Stephen P.(1-BYU)
Cogrowth of groups and the Dedekind-Frobenius group determinant. 
Math. Proc. Cambridge Philos. Soc. 121 (1997), no. 2, 193–217


I am not sure it is as general as you might like, but...


