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

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 generated by the set $$S^G=\{s^g : g\in G\} = \{g^{-1}sg : g\in G\}$$ of all conjugates of elements of $$S$$ in $$G.$$

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

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

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 or, defined as the join of all normal subgroups contained in $$S.$$

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 $$S.$$