TITLE: Determinant of Cayley table
QUESTION [5 upvotes]: Let $G$ be a finite group of order $n$ and $kG$ the group ring over a field of characteristic $0$. Let $C$ denote the Cayley table. The determinant of $C$ in $kG$ is defined as $det(C)=\sum_{\sigma \in S_n}sgn(\sigma)\, C_{1,\sigma(1)}\cdot ... \cdot C_{n,\sigma(n)}$. Does this always vanish? Clearly, $C$ is equivalent to the 'skew-symmetric' matrix $\tilde{C}_{g,h}=gh^{-1}$, but still I have no good argument.. but maybe it's wrong, I checked it only until $n=4$..

REPLY [3 votes]: You might want to read a fantastic overview of the theory of representIons of finite groups by prof T. Y. Lam: http://www.ams.org/notices/199803/lam.pdf
Look for the group determinant.