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]]. | ||
Revision as of 22:44, 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.