Deficiency of a group

This article defines an arithmetic function on groups
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Definition

Symbol-free definition

The deficiency of a finitely presented group is the maximum over all its finite presentations, of the deficiency of each presentation. In other words, it is the maximum possible value we can obtain for number of generators minus number of relations.

Note that in some conventions, deficiency of a presentation is defined as the number of relations minus the number of generators. In this case, the deficiency of a group is defined as the minimum of the deficiencies of all its presentations, and this notion of deficiency is the negative of the notion of deficiency defined above.