Normal closure

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

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 (join) 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

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.

Related subgroup properties

Subgroup property	Definition in terms of normal closure	Abstraction/additional comment
normal subgroup	equals its own normal closure; equivalently, arises as the normal closure of something	the normal closure operator is idempotent, so its image space and fixed point space coincide.
contranormal subgroup	normal closure is whole group	the inverse image of the whole group under the normal closure operator is the set of contranormal subgroups
2-subnormal subgroup	normal in its normal closure	the fixed point space and image space of the operator obtained by applying normal closure in normal closure coincide, and both equal 2-subnormality
subnormal subgroup	applying normal closure in (i.e., inner iteration) enough times gets to the subgroup	A $k$-subnormal subgroup is a subgroup that equals its normal closure in normal closure in ... where normal closure occurs $k$ times.
subgroup whose normalizer equals its normal closure	normal closure equals normalizer
NE-subgroup	subgroup equals intersection of its normalizer and normal closure
nearly normal subgroup	subgroup of finite index in its normal closure
closure-characteristic subgroup	normal closure is a characteristic subgroup

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.

References

Textbook references

• A Course in the Theory of Groups by Derek J. S. Robinson, ISBN 0387944613, ^{More info}, Page 16, Normal closure and core

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Definition links

• Wikipedia page