Nilpotency of fixed class is quotient-closed

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

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

Related facts

• Nilpotency of fixed class is subgroup-closed
• Nilpotency of fixed class is varietal
• Nilpotency of fixed class is direct product-closed