12.79 (1992)

Suppose that $a$ and $b$ are two elements of a finite group $G$ such that the function $$\phi(g) = 1_G(g) - 1^G_{\langle a \rangle}(g) - 1^G_{\langle b \rangle}(g) - 1^G_{\langle ab \rangle}(g) + 2$$ is a character of $G$. Is it true that $G = \langle a, b \rangle$? The converse statement is true.