Epinilpotent group

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Definition

A group $G$ is termed epinilpotent if there is a nonnegative integer $c$ such that it satisfies the following equivalent conditions:

1. The class c-epicenter of $G$ equals the whole group $G$.
2. The only quotient group of $G$ that is a class c-capable group is the trivial group.

The smallest $c$ satisfying any of these equivalent conditions is termed the epinilpotency class.

Note that finitely generated and epinilpotent implies cyclic, so among finitely generated groups, the only epinilpotent group are cyclic, hence epabelian, so they have epinilpotency class one.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
epabelian group	epinilpotency class one
cyclic group		via epabelian
locally cyclic group		via epabelian
periodic divisible abelian group		via epabelian

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
nilpotent group