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 ![]() ![]() |
---|---|---|
1 | normal closure in the whole group of the subgroup generated by that subset | normal closure ![]() ![]() ![]() |
2 | smallest normal subgroup of the whole group that contains the subset | smallest subgroup ![]() ![]() ![]() ![]() |
3 | subgroup generated by the set of all conjugate elements to elements of the subset | subgroup ![]() ![]() |
4 | the unique smallest possible kernel of a homomorphism from the whole group whose kernel contains the subset | the smallest subgroup ![]() ![]() ![]() ![]() ![]() ![]() |
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.