A basic fact about perfect groups is Grün's lemma from (Grün 1935, Satz 4, p. 3): the quotient of a perfect group by its center is centerless (has trivial center). Proof: If G is a perfect group, let Z 1 and Z 2 denote the first two terms of the upper central series of G (i.e., Z 1 is the center of G, and Z 2 /Z 1 is the center of G/Z 1).
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