Definition of finite-by-nilpotent In group theory what are definitions of: 1. finite-by-nilpotent, 2. nilpotent-by-finite, 3. abelian-by-finite, 4. or in general: *-by- *?

A group $G$ is called X-by-Y if it has a normal subgroup $N$ that belongs to the class X and the quotient $G/N$ belongs to the class Y.