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

Given a finite group [math]H[/math], setting [math]A=C^*(H)[/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^{opp} [[/math]]

given by the formulae [math]\Delta(g)=g\otimes g[/math], [math]\varepsilon(g)=1[/math], [math]S(g)=g^{-1}[/math] and linearity, are subject to the same conditions as above, namely

[[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].