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 $\operatorname_G(S)$ of $S$ is the intersection of all normal subgroups of $G$ containing $S$:
$\operatorname_G(S) = \bigcap_ N.$

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

The subgroup $\operatorname_G(S)$ is the subgroup generated by the set $S^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:
$\operatorname_G(S) = \.$

Any normal subgroup is equal to its normal closure. The normal 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 $G = F(S) / \operatorname_(R),$ where $F(S)$ is a free group on $S.$

References

Group theory
Closure operators

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