Closing the Relation Gap by Direct Product Stabilization Original Research Article

The generation gap of a groupGis the difference between the minimal number of generators ofGand the rank of the augmentation ideal. The relation gap of a presentationF/Nis the difference between the minimal number of elements that generateNas a normal subgroup and the minimal number ofG-module generators of the relation moduleN/[N, N]. We show that ifGis a finitely presented group then there existsnsuch thatG×∏i=1n Zp,Zpbeing the cyclic group of orderp, has zero generation and zero relation gap. We apply this result to questions concerning the efficiency of finite groups.