Epinilpotent group
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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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Contents
Definition
A group is termed epinilpotent if there is a nonnegative integer
such that it satisfies the following equivalent conditions:
- The class c-epicenter of
equals the whole group
.
- The only quotient group of
that is a class c-capable group is the trivial group.
The smallest 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 |