Suppose $G$ is a non-Abelian group. $a$ an involution in $G$, and $b$ an element of order at least $3$. If for each (complex) representation $\tau$ of $G$, we know that $\tau(a+b+b^{-1})$ has rational eigenvalues, can someone find a bound on the size of the subgroup generated by $a$ and $b$, i.e $|\langle \{a,b\}\rangle|$? The bound should be a number less than or equal $25$, but I have no idea how to prove this!