Normal subgroup generated by a subset: Difference between revisions

Revision as of 15:47, 8 May 2014

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 $N\leq G$ 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 $G$ .

Related notions

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

@@ Line 10: / Line 10: @@
 | 1 || [[defining ingredient::normal closure]] in the whole group of the [[defining ingredient::subgroup generated by a subset|subgroup generated]] by that subset || normal closure <math>\langle A \rangle^G</math> where <math>\langle A \rangle</math> is the subgroup generated by <math>A</math>
 |-
-| 2 || smallest [[defining ingredient::normal subgroup]] of the whole group that contains the subset || smallest subgroup <math>N \le G</math> such that <math>A \le N</math> and <math>N</math> is normal in <math>G</math>
+| 2 || smallest [[defining ingredient::normal subgroup]] of the whole group that contains the subset || smallest subgroup <math>N \le G</math> such that <math>A \subseteq N</math> and <math>N</math> is normal in <math>G</math>
 |-
 |3 || [[subgroup generated by a subset|subgroup generated]] by the set of all conjugate elements to elements of the subset || subgroup <math>\langle B \rangle</math> where <math>B = \bigcup_{g \in G} gAg^{-1}</math>