# Efficient group

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