Closing the Relation Gap by Direct Product Stabilization

✍ Scribed by J. Harlander

• Book ID: 102569559
• Publisher: Elsevier Science
• Year: 1996
• Tongue: English
• Weight: 152 KB
• Volume: 182
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.1996.0184

✦ Synopsis

The generation gap of a group G is the difference between the minimal number of generators of G and the rank of the augmentation ideal. The relation gap of a presentation FrN is the difference between the minimal number of elements that generate N as a normal subgroup and the minimal number of G-module generaw x tors of the relation module Nr N, N . We show that if G is a finitely presented group then there exists n such that G = Ł n Z , Z being the cyclic group of is 1 p p order p, has zero generation and zero relation gap. We apply this result to questions concerning the efficiency of finite groups.