This is a big-list question that I suppose could be a community wiki post. If this is too broad or otherwise a bad question, I'm sorry.
The Question:
Let $S\subseteq G$ for a group $G$. What is the relationship between $\langle S\rangle_G$ and ${\rm ncl}_G(S)$?
The Details:
Here:
- $\langle S\rangle_G$ is the subgroup of $G$ generated by $S$. It could be defined as $$\langle S\rangle_G=\bigcap_{H\le G,\\ S\subseteq H}H.$$
- ${\rm ncl}_G(S)$ is the normal subgroup of $G$ generated by $S$. It can be defined as $${\rm ncl}_G(S)=\bigcap_{N\unlhd G,\\ S\subseteq N}N.$$ Another name for this is the normal closure of $S$.
Context:
I'm looking at a function $\Delta$ of groups defined in terms of normally generating sets. That is, I'm interested in $T\subseteq G$ for a group $G$ such that
$${\rm ncl}_G(T)=G.$$
It occurred to me that generating sets might be of some use. That is, I'm also interested in $U\subseteq G$ for a group $G$ such that
$$\langle U\rangle_G=G.$$
What is the interplay between the $T$s and the $U$s?
The best I could find was . . .
The normal subgroup generated by a subset depends on the ambient group, unlike the subgroup generated by a subset. In other words, if $A$ is a subset of a group $H$ which is a subgroup of a group $G$, the normal subgroup generated by $A$ in $H$ may differ from the normal subgroup generated by $A$ in $G$.
. . . here.
What kind of answers am I looking for?
A list of theorems relating $\langle S\rangle_G$ to ${\rm ncl}_G(S)$. Proofs and references are optional.
Since this is a big-list question, I will not accept an answer, unless it is somehow provably exhaustive.
Appendices:
As per this, I thought I'd add more details in these appendices.
Appendix (Definition of $\Delta$):
Define:
- ${\rm Conj}_G(X^{\pm 1})=\{gxg^{-1}, gx^{-1}g^{-1}\mid x\in X,g\in G\}$ for $X\subseteq G$.
- $\| g\|_S=\inf\{n\in\Bbb N\mid g=x_1\dots x_n, x_i\in{\rm Conj}_G(S^{\pm 1})\}$ for $g\in G$; and if no such $n$ exists, let $\|g\|_S=\infty$.
- $\| G\|_S=\sup\{\| g\|_S\mid g\in G\}$.
- $\Gamma(G)=\{ S\subseteq G\mid |S|<\infty, {\rm ncl}_G(S)=G\}$.
Then
$$\Delta(G)=\sup\{\| G\|_S\mid S\in \Gamma(G)\}.$$
Appendix 2 (Further Motivation):
I have some groups $G$ I would like to know $\Delta(G)$ for. I have some nice presentations and generating sets for them. My hope is that I could use those sets to bound $\Delta(G)$.
Let $G$ be any group, $T$ be any subset of $G$, and $$T^G=\{g^{-1}tg: t\in T, g\in G\}.$$ It is easy to check that ${\rm ncl}_G(T)=G$ iff $\langle T^G\rangle_G=G$.