Editing Normal closure (group theory)

{{short description|Smallest normal group containing a set}}
{{About|the normal closure of a subset of a group|the normal closure of a field extension|Normal closure (field theory)}}
{{Group theory sidebar}}
In [[group theory]], the '''normal closure''' of a [[subset]] <math>S</math> of a [[Group (mathematics)|group]] <math>G</math> is the smallest [[normal subgroup]] of <math>G</math> containing <math>S.</math>

== Properties and description ==

Formally, if <math>G</math> is a group and <math>S</math> is a subset of <math>G,</math> the normal closure <math>\operatorname{ncl}_G(S)</math> of <math>S</math> is the intersection of all normal subgroups of <math>G</math> containing <math>S</math>:<ref name=HEOB>{{cite book|title=Handbook of Computational Group Theory|author=Derek F. Holt|author2=Bettina Eick|author3=Eamonn A. O'Brien|publisher=CRC Press|year=2005|isbn=1-58488-372-3|page=[https://archive.org/details/handbookofcomput0000holt/page/14 14]|url=https://archive.org/details/handbookofcomput0000holt/page/14}}</ref>
<math display="block">\operatorname{ncl}_G(S) = \bigcap_{S \subseteq N \triangleleft G} N.</math>

The normal closure <math>\operatorname{ncl}_G(S)</math> is the smallest normal subgroup of <math>G</math> containing <math>S,</math><ref name="HEOB" /> in the sense that <math>\operatorname{ncl}_G(S)</math> is a subset of every normal subgroup of <math>G</math> that contains <math>S.</math>

The subgroup <math>\operatorname{ncl}_G(S)</math> is the subgroup [[Generating set of a group|generated]] by the set <math>S^G=\{s^g : s \in S, g\in G\} = \{g^{-1}sg : s \in S, g\in G\}</math> of all [[Conjugacy class|conjugates]] of elements of <math>S</math> in <math>G.</math>
Therefore one can also write the subgroup as the set of all products of conjugates of elements of <math>S</math> or their inverses:
<math display="block">\operatorname{ncl}_G(S) = \{g_1^{-1}s_1^{\epsilon_1} g_1\cdots g_n^{-1}s_n^{\epsilon_n}g_n : n \geq 0, \epsilon_i = \pm 1, s_i\in S, g_i \in G\}.</math>

Any normal subgroup is equal to its normal closure. The normal closure of the [[empty set]] <math>\varnothing</math> is the [[trivial subgroup]].<ref>{{cite book|last1=Rotman|first1=Joseph J.|title=An introduction to the theory of groups|series=Graduate Texts in Mathematics|date=1995|volume=148|publisher=[[Springer-Verlag]]|location=New York|isbn=0-387-94285-8|page=32|edition=Fourth|url=https://books.google.com/books?id=7-bBoQEACAAJ|mr=1307623|doi=10.1007/978-1-4612-4176-8}}</ref>

A variety of other notations are used for the normal closure in the literature, including <math>\langle S^G\rangle,</math> <math>\langle S\rangle^G,</math> <math>\langle \langle S\rangle\rangle_G,</math> and <math>\langle\langle S\rangle\rangle^G.</math>

Dual to the concept of normal closure is that of {{em|normal interior}} or {{em|[[normal core]]}}, defined as the join of all normal subgroups contained in <math>S.</math><ref>{{cite book|title=A Course in the Theory of Groups|volume=80|series=Graduate Texts in Mathematics|first=Derek J. S.|last=Robinson|publisher=[[Springer-Verlag]]|year=1996|isbn=0-387-94461-3|zbl=0836.20001|edition=2nd|page=16 }}</ref>

== Group presentations ==

For a group <math>G</math> given by a [[Presentation of a group|presentation]] <math>G=\langle S \mid R\rangle</math> with generators <math>S</math> and defining [[relator]]s <math>R,</math> the presentation notation means that <math>G</math> is the [[quotient group]] <math>G = F(S) / \operatorname{ncl}_{F(S)}(R),</math> where <math>F(S)</math> is a [[free group]] on <math>S.</math><ref> 
{{cite book|last1=Lyndon|first1=Roger C.|author1-link=Roger Lyndon|last2=Schupp|first2=Paul E.|authorlink2=Paul Schupp|isbn=3-540-41158-5|mr=1812024|page=87|publisher=Springer-Verlag, Berlin|series=Classics in Mathematics|title=Combinatorial group theory|url=https://books.google.com/books?id=cOLrCAAAQBAJ|year=2001}}
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== References ==

{{reflist}}

[[Category:Group theory]]
[[Category:Closure operators]]


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