Let $G$ be a group and let $A$ be the subgroup of $G$ generated by $\{a_i\}_{i\in I}$; let $B$ be the subgroup of $G$ generated by $\{b_j\}_{j\in J}$, where $I$ and $J$ are index sets.
Is there a way to find a generating set of the subgroup $A\bigcap B$; or a general expression of an arbitrary element in $A\bigcap B$?
If the answer is negative, under what condition on $G$ (or on $A$ and $B$) where there is a nice result?