Normal closure

Symbol-free definition
The normal closure of a subgroup in a group can be defined in any of the following equivalent ways:


 * As the intersection of all defining ingredient::normal subgroups containing the given subgroup
 * As the subgroup generated (join) by all defining ingredient::conjugate subgroups to the given subgroup
 * As the set of all elements that can be written as products of finite length of elements from the subgroup and their conjugates
 * As the kernel of the smallest homomorphism from the whole group which annihilates the given subgroup

Normal closure is also used for normal closure of a subset which is also termed the normal subgroup generated by a subset. The normal subgroup generated by a subset is defined as the normal closure of the subgroup generated by that subset.''

Definition with symbols
The normal closure of a subgroup $$H$$ in a group $$G$$, denoted as $$H^G$$ is defined in the following equivalent ways:


 * As the intersection of all normal subgroups of $$G$$ containing $$H$$
 * As the subgroup generated by all $$gHg^{-1}$$ where $$gHg^{-1}$$ denotes a conjugate of $$H$$ by $$g$$.

Normal closure, when used for subsets of a group, means the normal subgroup generated by the subset.

Image
The normal closure operator is an idempotent operator (viz the normal closure of the normal closure is again the normal closure) and the fixed-point-cum-image subgroups are precisely the normal subgroups. In other words, the normal closure of any subgroup is a normal subgroup, and the normal closure of a normal subgroup is itself.

Inverse image of whole group
A subgroup whose normal closure is the whole group is termed contranormal.

In general, no proper subnormal subgroup can be contranormal.

Inner iteration
The $$k$$-times inner iteration of the normal closure denotes the $$k$$-subnormal closure of the subgroup. This is again an idempotent operator and the fixed-point cum image space is precisely the space of $$k$$-subnormal subgroups.

Computation
The normal closure of a subgroup in a group can be found computationally by invoking the membership testing problem. It is a variant of the normality testing problem.

Textbook references

 * , Page 16, Normal closure and core