Epabelian group: Difference between revisions
(Created page with "{{group property}} ==Definition== A group <math>G</math> is termed '''epabelian''' if it satisfies the following equivalent conditions: # For any group <math>K</math> w...") |
No edit summary |
||
| Line 27: | Line 27: | ||
| [[Stronger than::epinilpotent group]] || || || || | | [[Stronger than::epinilpotent group]] || || || || | ||
|} | |} | ||
===Collapse=== | |||
Any finitely generated epabelian group is cyclic. This follows from the more general fact that [[finitely generated and epinilpotent implies cyclic]]. | |||
Revision as of 22:31, 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 .
Relation with other properties
Stronger properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| cyclic group |
Weaker properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| abelian group | ||||
| epinilpotent group |
Collapse
Any finitely generated epabelian group is cyclic. This follows from the more general fact that finitely generated and epinilpotent implies cyclic.