# Epinilpotent group

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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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