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 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>:
<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>
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 or , defined as the join of all normal subgroups contained in <math>S.</math>
Group presentations
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>