# Normal subgroup generated by a subset

## Definition

### 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:

No. The normal subgroup generated by a subset is the ... The normal subgroup generated by a subset $A$ of a group $G$ is the ...
1 normal closure in the whole group of the subgroup generated by that subset normal closure $\langle A \rangle^G$ where $\langle A \rangle$ is the subgroup generated by $A$
2 smallest normal subgroup of the whole group that contains the subset smallest subgroup $N \le G$ such that $A \subseteq N$ and $N$ is normal in $G$
3 subgroup generated by the set of all conjugate elements to elements of the subset subgroup $\langle B \rangle$ where $B = \bigcup_{g \in G} gAg^{-1}$
4 the unique smallest possible kernel of a homomorphism from the whole group whose kernel contains the subset the smallest subgroup $N$ containing $A$ for which there is a homomorphism $\varphi:G \to K$ such that the kernel of $\varphi$ equals $N$. Any other subgroup arising as such a kernel must contain $N$.

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 $G$.