Normal closure (group theory)

{{About|the normal closure of a subset of a group|the normal closure of a field extension|Normal closure (field theory)}}

In group theory, the normal closure of a subset $S$ of a group $G$ is the smallest normal subgroup of $G$ containing $S.$

Properties and description

Formally, if $G$ is a group and $S$ is a subset of $G,$ the normal closure $\operatorname{ncl}_G(S)$ of $S$ is the intersection of all normal subgroups of $G$ containing $S$ :{{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}}

$\operatorname{ncl}_G(S) = \bigcap_{S \subseteq N \triangleleft G} N.$

The normal closure $\operatorname{ncl}_G(S)$ is the smallest normal subgroup of $G$ containing $S,$ in the sense that $\operatorname{ncl}_G(S)$ is a subset of every normal subgroup of $G$ that contains $S.$

The subgroup $\operatorname{ncl}_G(S)$ is the subgroup generated by the set $S^G=\{s^g : s \in S, g\in G\} = \{g^{-1}sg : s \in S, g\in G\}$ of all conjugates of elements of $S$ in $G.$

Therefore one can also write the subgroup as the set of all products of conjugates of elements of $S$ or their inverses:

$\operatorname{ncl}_G(S) = \{g_1^{-1}s_1^{\epsilon_1} g_1\cdots g_n^{-1}s_n^{\epsilon_n}g_n : n \geq 0, \epsilon_i = \pm 1, s_i\in S, g_i \in G\}.$

Any normal subgroup is equal to its normal closure. The normal closure of the empty set $\varnothing$ is the trivial subgroup.{{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}}

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

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 $S.$ {{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 }}

Group presentations

For a group $G$ given by a presentation $G=\langle S \mid R\rangle$ with generators $S$ and defining relators $R,$ the presentation notation means that $G$ is the quotient group $G = F(S) / \operatorname{ncl}_{F(S)}(R),$ where $F(S)$ is a free group on $S.$

{{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}}

References

Category:Group theory

Category:Closure operators