In group theory, the normal closure of a subset of a group is the smallest normal subgroup of containing
Properties and description
Formally, if is a group and is a subset of the normal closure of is the intersection of all normal subgroups of containing :
The normal closure is the smallest normal subgroup of containing in the sense that is a subset of every normal subgroup of that contains
The subgroup is generated by the set of all conjugates of elements of in
Therefore one can also write
Any normal subgroup is equal to its normal closure. The conjugate closure of the empty set is the trivial subgroup.
A variety of other notations are used for the normal closure in the literature, including and
Dual to the concept of normal closure is that of or , defined as the join of all normal subgroups contained in
Group presentations
For a group given by a presentation with generators and defining relators the presentation notation means that is the quotient group where is a free group on