Efficient group: Difference between revisions
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==Definition== | ==Definition== | ||
A [[finitely presented group]] is said to be '''efficient''' if its [[deficiency of a group|deficiency]] equals the [[rank of a group|rank]] of its [[Schur multiplier]]. In other words, it possesses a finite [[presentation of a group|presentation]] where the number of relations equals the number of generators plus the rank of the Schur multiplier. | A [[finitely presented group]] is said to be '''efficient''' if its [[defining ingredient::deficiency of a group|deficiency]] equals the negative of the [[rank of a group|rank]] of its [[defining ingredient::Schur multiplier]]. In other words, it possesses a finite [[defining ingredient::presentation of a group|presentation]] where the number of relations equals the number of generators plus the rank of the Schur multiplier. Such a presentation is termed an [[defining ingredient::efficient presentation]]. The term is typically used for [[finite group]]s. | ||
==Relation with other properties== | |||
===Stronger properties=== | |||
{| class="wikitable" border="1" | |||
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | |||
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| [[Weaker than::Finite cyclic group]] || || || || | |||
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| [[Weaker than::Group with zero deficiency]] || || || || | |||
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| [[Weaker than::Finite abelian group]] || || || || | |||
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Latest revision as of 22:53, 14 April 2010
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 finitely presented group is said to be efficient if its deficiency equals the negative of the rank of its Schur multiplier. In other words, it possesses a finite presentation where the number of relations equals the number of generators plus the rank of the Schur multiplier. Such a presentation is termed an efficient presentation. The term is typically used for finite groups.
Relation with other properties
Stronger properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| Finite cyclic group | ||||
| Group with zero deficiency | ||||
| Finite abelian group |