Efficient group

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This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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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