Normal closure

Let $𝐺$ be a group, and $𝑆 \subseteq 𝐺$ be a subset. The normal closure $n c l 𝐺 (𝑆)$ of $𝑆$ is the smallest normal subgroup of $𝐺$ containing $𝑆$ . #m/def/group This is well-defined because the intersection of normal subgroups is a normal subgroup, hence

n c l 𝐺 (𝑆) = ⋂ {𝑁 ⊴ 𝐺 : 𝑆 \subseteq 𝑁}

Properties

$n c l 𝐺 (𝑆)$ has the conjugates of elements of $𝑆$ as generators.

#state/tidy | #lang/en | #SemBr