Centralizer of derived subgroup has class at most two

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Suppose G is a group and H = C_G([G,G]) is the Centralizer of commutator subgroup (?) of G. Then, H is a Group of nilpotency class two (?). In other words, H is either an abelian group or a non-abelian group of class two. In particular, H is a Class two normal subgroup (?) and a Class two characteristic subgroup (?).

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Given: A group G. H = C_G([G,G]).

To prove: H is a group of class (at most) two.

Proof: Since [G,G] centralizes H, and H \le G, [H,H] centralizes H. Thus, [H,H] is contained in the center of H. Thus, H has class at most two.