[math] \newcommand{\mathds}{\mathbb}[/math]

Given a finite group [math]G[/math], setting [math]A=C(G)[/math], prove that the maps

[[math]] \Delta:A\to A\otimes A [[/math]]

[[math]] \varepsilon:A\to\mathbb C [[/math]]

[[math]] S:A\to A [[/math]]

which are transpose to the multiplication [math]m:G\times G\to G[/math], unit [math]u:\{.\}\to G[/math] and inverse map [math]i:G\to G[/math], are subject to the following conditions

[[math]] (\varepsilon\otimes id)\Delta=(id\otimes\varepsilon)\Delta=id [[/math]]

[[math]] m(S\otimes id)\Delta=m(id\otimes S)\Delta=\varepsilon(.)1 [[/math]]

in usual tensor product notation, along with the extra condition [math]S^2=id[/math].