In section 2.5 of his Linear representations of finite groups (I have the french copy), Serre gives an example of determination of the character table of a group $G$. The group $G$ is taken to be $S_3$. He points out that the order of $G$ is $6$, that there are three conjugacy classes (the identity $1$, the three transpositions and the two cyclic permutations, which imply there are exactly three irreducbile characters), and that if $t$ is a transposition and $c$ a cyclic permutation, then $t^2=1$, $c^3=1$ and $tc=c^2t$. No problem until here. But then he says
Hence we have two irreducible characters of degree one: the unit character (character of the unitary representation) and the character giving the signature of a permutation.
What I would like to understand is how the above imply the quoted part, since the argument seems to come up again in the example of the following section. Note that he doesnt directly define the character as a homomorphism, which would have made the answer almost obvious to me, but as a trace (if $\rho$ is a reperesenation with character $\chi$, then $\chi (s)=Tr(\rho_s)$ ).