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, 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 amore or less directattempt 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?