# Efficient group

From Groupprops

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 propertiesVIEW 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 |