# Deficiency of a presentation

From Groupprops

## Definition

### Symbol-free definition

The **deficiency** of a presentation of a group is the number of generators minus the number of relations in a presentation.

(Note that in some conventions, it is defined as the number of relations minus the number of generators. All the deficiency values would thus get negated and all occurrences of *minimum* would get replaced by *maximum* and vice versa).

Note that since trivial and redundant relations can be thrown into any presentation, the deficiency of a presentation can be pushed to as low as we please. The challenge is in obtaining a presentation with high deficiency. The deficiency of a group is the maximum possible deficiency over all presentations of a group.