Normal subgroup generated by a subset

Tabular definition
The normal subgroup generated by a subset, sometimes also called the normal closure of a subset, is defined in the following equivalent ways:

The normal subgroup generated by a subset $$A$$ of a group $$G$$ is denoted $$\langle A^G \rangle$$, $$\langle A \rangle^G$$, or sometimes simply as $$A^G$$, though the final notation may also be used simply for the union of conjugates of $$A$$.

Facts
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 $$G$$.

Related notions

 * Normal closure of finite subset is a subgroup that arises as the normal subgroup generated by a finite subset.