Efficient group - Revision history

Vipul at 22:53, 14 April 2010

2010-04-14T22:53:32Z

← Older revision		Revision as of 22:53, 14 April 2010
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	==Definition==		==Definition==

	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]].		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
			\|-
			\| [[Weaker than::Finite cyclic group]] \|\| \|\| \|\| \|\|
			\|-
			\| [[Weaker than::Group with zero deficiency]] \|\| \|\| \|\| \|\|
			\|-
			\| [[Weaker than::Finite abelian group]] \|\| \|\| \|\| \|\|
			\|}

Vipul at 22:44, 14 April 2010

2010-04-14T22:44:16Z

← Older revision		Revision as of 22:44, 14 April 2010
Line 3:		Line 3:
	==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]].

Vipul: 1 revision

2008-05-07T23:26:23Z

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Vipul at 12:51, 15 May 2007

2007-05-15T12:51:39Z

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{{group property}}

==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.