If$\ G$ is a group, and the set $S=\{a,b\}$ is a subset of $\ G$, can we say that the smallest subgroup of $\ G$ generated by $\langle a,b\rangle$ will always be either $\langle a \rangle$, $\langle b\rangle$, or in the case that $\langle a\rangle$ does not generate $b$ and $\langle b\rangle$ does not generate $a$, then $\langle a,b\rangle$ = $G$?
I'm having difficulty trying to think of a counterexample to this, particularly a finite group whose elements can be easily enumerated (e.g. $\ U(n)$ the multiplicative group of integers modulo $n$).