In the theory of rings we have clear descriptions of the smallest ideal in a ring $R$ containing a subset $S$ of $R$. I'd like to know if there is such a description in group theory, that is, if $N(S)$ is the smallest normal subgroup of $G$ containing $S$, can you describe $N(S)$ using the operations of groups and the elements of both $S$ and $G$? Is there such description if $S$ is a subgroup? Thanks.