Ex 2.6 Character Theory, Isaac: Let $\chi$ be a charcter of $G$, such that $\chi(1)>1$. Show that $\chi \bar{\chi} \notin Irr(G)$. $Irr(G)$ denotes the set of characters induced from irreducible representations.
What I can deduce so far...
(i) If $\chi$ were a linear character, and $\chi \in Irr(G)$, then I can prove that the product is in fact in $Irr(G)$.
(ii) If $\chi \bar{\chi} = |\chi|^2 \in Irr(G)$, then by orthogonality relations, $[|\chi|^2, |\chi|^2] =1$, i.e. $\frac{1}{|G|} \sum_{g \in G} |\chi(g)|^4 =1.$