Minimal presentations for groups of order 2ⁿ, n ≤ 6

Abstract

Let G be a finite 2-group having a minimal generating set {x1, …, xr} so that r = d (G) is an invariant of G. Suppose further that G has a presentation then.

On equality of central and class preserving automorphisms of finite p-groups

“…We denote the ith family by Φ i . As mentioned in [16], the groups are listed in same order in both [8] and [16] and so we take the liberty of choosing the presentation of a group from any of the lists. Unless or otherwise stated, we use Sag and Wamsley's list and adopt the same notations for the nomenclature and presentations of the groups.…”

Class-preserving automorphisms of some finite p-groups

“…Sag and Wamsley [10] have given minimal presentations of these groups. As mentioned in [10], the groups are in the same order in [4] and [10]. We denote the i-th group as G i .…”

“…The last family contains 2-generated groups G 49 −G 51 of maximal class. It is clear from [10] that in each of these groups, one generator, say x, is of order 16 and hence |x G | = 2. If y is another generator, then y −1 xy is respectively x 15 , x 7 and x 15 in the group G 49 , G 50 and G 51 .…”

“…He then showed, by picking up one group from each isoclinism family, that Out c (G) = 1 for the groups 7 (1 5 ) and 10 (1 5 ) from seventh and tenth family in [6], and hence concluded that if G is a finite p-group of order p 5 , where p is an odd prime, then Out c (G) = 1 if and only if G is isoclinic to a group either in the seventh or in the tenth family. Recently, Kalra and Gumber (Theorem 4.2 of [7]), using the classifications, isoclinism families and presentations given by Hall and Senior [4] and Sag and Wamsley [10], have shown that if G is a finite 2-group of order 2 5 , then Out c (G) = 1 except for the forty fourth and forty fifth groups from the sixth family in [4].…”

Finite groups of arbitrary deficiency

“…A search of the literature on deficiencies of finite groups suggests that one can extract examples as needed by Kotschick from the work of Sag and Wamsley, who claimed to have computed the deficiency of every group of order for . However, they did not publish proofs, and the article does contain a number of errors beyond the obvious misprints: some presentations are not efficient as claimed, and others do not define the groups they should.…”