Template:Short description Template:About Template:Group theory sidebar In group theory, the normal closure of a subset <math>S</math> of a group <math>G</math> is the smallest normal subgroup of <math>G</math> containing <math>S.</math>
Properties and descriptionEdit
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>Template:Cite book</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 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 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>Template:Cite book</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 Template:Em or Template:Em, defined as the join of all normal subgroups contained in <math>S.</math><ref>Template:Cite book</ref>
Group presentationsEdit
For a group <math>G</math> given by a presentation <math>G=\langle S \mid R\rangle</math> with generators <math>S</math> and defining relators <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> Template:Cite book </ref>