Normal closure

This article defines a subgroup operator related to the subgroup property normality. By subgroup operator is meant an operator that takes as input a subgroup of a group and outputs a subgroup of the same group.

This article is about a standard (though not very rudimentary) definition in group theory. The article text may, however, contain more than just the basic definition
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Definition

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 normal subgroups containing the given subgroup
• As the subgroup generated by all 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

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 $g$.

Related subgroup properties

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

BEWARE! This section of the article uses terminology local to the wiki, possibly without giving a full explanation of the terminology used (though efforts have been made to clarify terminology as much as possible within the particular context)

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

Further information: normal closure-finding

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.