Groupprops, The Group Properties Wiki (pre-alpha)
TIP: Beware of terminology local to the wiki
ABOUT US: We use a Creative Commons license. All our content is free to reuse, with attribution. Learn more
ALSO CHECK OUT: Topospaces: The Topology Wiki
Normal closure
From Groupprops
|
This article is about a standard (though not very rudimentary) definition in group theory.[SHOW MORE]
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
The normal closure of 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 HG 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.
The normal closure of a subset A of G is defined as the normal closure of the subgroup 
of 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.
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
External links
Search for "normal+closure" on the World Wide Web:
Scholarly articles: Google Scholar, JSTOR
Books: Google Books, Amazon
This wiki: Internal search, Google site search
Encyclopaedias: Wikipedia (or using Google), Citizendium
Math resource pages:Mathworld, Planetmath, Springer Online Reference Works
Math wikis: Topospaces, Diffgeom, Commalg, Noncommutative
Discussion fora: Mathlinks, Google Groups
The web: Google, Yahoo, Windows Live
Learn more about using the Searchbox OR provide your feedback
Definition links
