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.
[Math] Given a set $S$ in a group $G$, how does the smallest normal subgroup containing $S$ look like
abstract-algebragroup-theory
Best Answer
The smallset subgroup of $G$ containing the set $S$ is simply $\langle S \rangle$, the subgroup generated by all elements of $S$, as any smaller subgroup containing $S$ would not be closed.
To describe the smallest normal subgroup containing $S$, we do something similar. We want for $s^g$ to remain in the group for any $s\in S, g\in G$, so we define $S^G=\{s^g:s\in S, g\in G\}$. Then $\langle S^G \rangle$ is precisely what we want: the smallest subgroup of $G$ containing $S$ and all conjugates thereof. As you can see there is no difference in these definitions if $S$ is a subgroup.
This actually has a name: $\langle S^G \rangle$ is the normal closure, or conjugate closure, of $S$ in $G$. A bit more information on normal closures can be found here, though not much can be said about them in general without looking at specific cases.