Editing Perfect group (section)

==Grün's lemma==
A basic fact about perfect groups is '''Grün's lemma''' {{harv|Grün|1935|loc=Satz 4,<ref group="note">''[[wikt:Satz#German|Satz]]'' is German for "theorem".</ref> p.&thinsp;3}}, due to [[Otto Grün]]: the [[quotient group|quotient]] of a perfect group by its [[center (group theory)|center]] is centerless (has trivial center).

<blockquote>'''Proof:''' If ''G'' is a perfect group, let ''Z''<sub>1</sub> and ''Z''<sub>2</sub> denote the first two terms of the [[Central series#Upper central series|upper central series]] of ''G'' (i.e., ''Z''<sub>1</sub> is the center of ''G'', and ''Z''<sub>2</sub>/''Z''<sub>1</sub> is the center of ''G''/''Z''<sub>1</sub>). If ''H'' and ''K'' are subgroups of ''G'', denote the [[commutator]] of ''H'' and ''K'' by [''H'', ''K''] and note that [''Z''<sub>1</sub>, ''G''] = 1 and [''Z''<sub>2</sub>, ''G''] ⊆ ''Z''<sub>1</sub>, and consequently (the convention that [''X'', ''Y'', ''Z''] = [[''X'', ''Y''], ''Z''] is followed):

:<math>[Z_2,G,G]=[[Z_2,G],G]\subseteq [Z_1,G]=1</math>
:<math>[G,Z_2,G]=[[G,Z_2],G]=[[Z_2,G],G]\subseteq [Z_1,G]=1.</math>

By the [[three subgroups lemma]] (or equivalently, by the [[Commutator#Identities (group theory)|Hall-Witt identity]]), it follows that [''G'', ''Z''<sub>2</sub>] = [[''G'', ''G''], ''Z''<sub>2</sub>] = [''G'', ''G'', ''Z''<sub>2</sub>] = {1}. Therefore, ''Z''<sub>2</sub> ⊆ ''Z''<sub>1</sub> = ''Z''(''G''), and the center of the quotient group ''G'' / ''Z''(''G'') is the [[trivial group]].</blockquote>

As a consequence, all [[Center (group theory)#Higher centers|higher centers]] (that is, higher terms in the [[upper central series]]) of a perfect group equal the center.