Commutator of a group and a subgroup implies normal

Statement with symbols
Suppose $$G$$ is a group and $$H$$ is any subgroup. Then, the commutator $$[G,H]$$, defined as:

$$[G,H] := \langle [g,h] \mid g \in G, h \in H \rangle$$

is a normal subgroup of $$G$$.

Stronger facts

 * Commutator of a group and a subset implies normal
 * Subgroup normalizes its commutator with any subset
 * Commutator of a group and a subgroup of its automorphism group is normal

Other related facts

 * Commutator of two subgroups is normal in join
 * Commutator of a normal subgroup and a subset implies 2-subnormal
 * Commutator of a 2-subnormal subgroup and a subset implies 3-subnormal
 * Normality is commutator-closed
 * Characteristicity is commutator-closed

Breakdown for Lie rings

 * Lie bracket of a Lie ring and a subring need not be an ideal

Facts used

 * 1) uses::Subgroup normalizes its commutator with any subset: If $$K \le G$$ and $$A$$ is a subset of $$G$$, then $$K$$ normalizes the commutator $$[A,K] = [K,A]$$.

Proof
Given: A group $$G$$, a subgroup $$H$$.

To prove: The subgroup $$[G,H]$$ is normal in $$G$$.

Proof: Apply fact (1) to $$K = G$$ and $$A = H$$. We get that $$G$$ normalizes $$[G,H]$$. Hence, $$[G,H]$$ is normal in $$G$$.