Commutator map is homomorphism if commutator is in centralizer

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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

Facts used

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

Proof

The proof follows directly from Fact (1).