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 $⟨A{⟩}^G$ where $⟨A⟩$ 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 $⟨B⟩$ where $B={\bigcup }_{g\in G}gA{g}^{-1}$
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 $\phi :G\to K$ such that the kernel of $\phi$ 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 $⟨A^G⟩$, $⟨A{⟩}^G$, 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.