# Nilpotency of fixed class is quotient-closed

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This article gives the statement, and possibly proof, of a group property (i.e., nilpotent group) satisfying a group metaproperty (i.e., quotient-closed group property)

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## Statement

Suppose is a nilpotent group and is a normal subgroup of and is the corresponding quotient group. Then, is also a nilpotent group and its nilpotency class is at most equal to the nilpotency class of .

Note that we say that a group is nilpotent "of class " if its nilpotency class is at *most* . The statement can thus be reformulated as saying that the property of being nilpotent of class is closed under taking quotients.