Normal subgroup generated by a subset
From Groupprops
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 of a group is the ... |
---|---|---|
1 | normal closure in the whole group of the subgroup generated by that subset | normal closure where is the subgroup generated by |
2 | smallest normal subgroup of the whole group that contains the subset | smallest subgroup such that and is normal in |
3 | subgroup generated by the set of all conjugate elements to elements of the subset | subgroup where |
4 | the unique smallest possible kernel of a homomorphism from the whole group whose kernel contains the subset | the smallest subgroup containing for which there is a homomorphism such that the kernel of equals . Any other subgroup arising as such a kernel must contain . |
The normal subgroup generated by a subset of a group is denoted , , or sometimes simply as , though the final notation may also be used simply for the union of conjugates of .
Facts
The normal subgroup generated by a subset depends on the ambient group, unlike the subgroup generated by a subset. In other words, if is a subset of a group which is a subgroup of a group , the normal subgroup generated by in may differ from the normal subgroup generated by in .
Related notions
- Normal closure of finite subset is a subgroup that arises as the normal subgroup generated by a finite subset.