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 in a group , denoted as is defined in the following equivalent ways:
- As the intersection of all normal subgroups of containing
- As the subgroup generated by all where denotes a conjugate of by .
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 -subnormal subgroup is a subgroup that equals its normal closure in normal closure in ... where normal closure occurs 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 -times inner iteration of the normal closure denotes the -subnormal closure of the subgroup. This is again an idempotent operator and the fixed-point cum image space is precisely the space of -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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