Closing the Relation Gap by Direct Product Stabilization

Abstract

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. ᮊ

Embeddings into efficient groups

“…We remark that the author showed in [15] that a finite group can be embedded into a finite efficient group. In fact, if K is finite, then K x F]( =1 Z p is efficient for / big enough and p a prime.…”

“…We conclude by remarking that a subgroup of finite index of an efficient group need not be efficient. It was shown in [15] that a finite group can be embedded into a finite efficient group. Since there are non-efficient finite groups (Swan's examples for instance), finite index does not preserve efficiency.…”

Embeddings into k-Efficient Groups

“…Adapting ideas of Harlander [5], we show that the situation for finite nonabelian groups is to some extent redeemed by the following theorem. …”

“…Many finite groups are known not to admit such presentations 9 12 . However, Harlander [5] has proved that for any finite group G and any prime p coprime to the order of G there is an integer m such that the direct product G × m i=1 C p admits an efficient 2-presentation. The above theorem shows that Harlander's result holds with C p replaced by an arbitrary 2-efficient prime-power group.…”

Presentations of finite simple groups: a computational approach

“…On the other hand, [Har,Cor. 5.4] states that, if p is any prime not dividing the order of a finite group G, then G × P is efficient for all sufficiently large elementary abelian pgroups P .…”