Commutator map is homomorphism if commutator is in centralizer

Statement

Suppose $G$ is a group, $H$ is a subgroup of $G$, and $x\in G$ is such that:

$[x,H]\subseteq C_{G}(H)$

Then, the map $H\to G$ given by:

$y\mapsto [x,y]$

is a homomorphism of groups from $H$ to $G$.

Related facts

• Class two implies commutator map is endomorphism is a special case where the statement applies globally.

Facts used

1. Formula for commutator of element and product of two elements

Proof

The proof follows directly from Fact (1).