normal closure

Given a group $G$ and a subset $S\subset G$, the **normal closure** of $S$ in $G$ is the smallest subgroup containing the set of all conjugates of elements in $S$, i.e. the subgroup generated by the set of all elements of the form $g^{-1} s g$, where $g\in G$ and $s\in S$. The normal closure is clearly a normal subgroup of $G$.

The normal closure, also called **conjugate closure** (see Wikipedia), should be distinguished from the normalizer of $S$ in $G$. In combinatorial group theory a group $G$ is presented via specifying a set $X$ of generators and a set $R$ of relations. This means that $G=F/N$ where $F$ is the free group generated by $X$ and $N$ is the normal closure of $R$ in $F$.