Epabelian group: Difference between revisions
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# The [[defining ingredient::epicenter]] of <math>G</math> equals <math>G</math>. | # The [[defining ingredient::epicenter]] of <math>G</math> equals <math>G</math>. | ||
# <math>G</math> is an [[abelian group]] and its [[Schur multiplier]] (which is necessarily equal to its [[Schur multiplier of abelian group is its exterior square|exterior square]]) is the [[trivial group]]. | # <math>G</math> is an [[abelian group]] and its [[Schur multiplier]] (which is necessarily equal to its [[Schur multiplier of abelian group is its exterior square|exterior square]]) is the [[trivial group]]. | ||
# (''Certainly necessary, not sure it is sufficient''): For any elements <math>a,b \in G</math>, either <math>\langle a,b \rangle</math> is cyclic or there exists a positive integer <math>n</math> and an element <math>c \in G</math> such that <math>nc = a</math> and <math>nb = 0</math>. | |||
==Relation with other properties== | ==Relation with other properties== | ||
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| [[Weaker than::locally cyclic group]] || || || || | | [[Weaker than::locally cyclic group]] || || || || | ||
|- | |- | ||
| [[Weaker than::divisible | | [[Weaker than::periodic divisible abelian group]] || || || || | ||
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Latest revision as of 23:02, 9 June 2012
This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
View a complete list of group properties
VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions
Definition
A group is termed epabelian if it satisfies the following equivalent conditions:
- For any group with a central subgroup such that the quotient group is isomorphic to , must be an abelian group.
- The epicenter of equals .
- is an abelian group and its Schur multiplier (which is necessarily equal to its exterior square) is the trivial group.
- (Certainly necessary, not sure it is sufficient): For any elements , either is cyclic or there exists a positive integer and an element such that and .
Relation with other properties
Stronger properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| cyclic group | ||||
| locally cyclic group | ||||
| periodic divisible abelian group |
Weaker properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| abelian group | ||||
| epinilpotent group |
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Any finitely generated epabelian group is cyclic. This follows from the more general fact that finitely generated and epinilpotent implies cyclic.