Some 3-Generator, 3-Relation Finite 2-Groups

Groups of Order 2048 with Three Generators and Three Relations

“…It is noted in [6] that there are no 3-generator 2-groups of order less than 2 9 having trivial Schur multiplier and there exist exactly two such groups of order 2 9 . Moreover in [4] we see that there are exactly eighteen 3-generator 2-groups of order In this paper, using computational methods we show that there are exactly seventy-eight 3-generator 2-groups of order 2 11 with trivial Schur multiplier. We then give 3-generator, 3-relation presentations for fortyeight of them proving that these groups all have deficiency zero.…”

“…2 , where 0 ≤ ℓ ≤ 1. All 3-generator groups of order 2 10 with trivial Schur multiplier are classified in [4]. By using GAP we see that there is no immediate descendant of order 2 11 of these eighteen groups of order 2 10 .…”

“…Then an attempt was made to find a subset S of R having three elements such that X|S defines G. In searching for generating triples for each group a small set of group elements was chosen by a knowledge of conjugacy classes and checked for generating triples. This technique was also used in [4] to determine deficiency zero presentations for all 3-generator, 2-groups of order 2 10 with trivial Schur multiplier. The authors obtained seventeen deficiency zero presentations from eighteen groups in [4] by this method.…”

“…This technique was also used in [4] to determine deficiency zero presentations for all 3-generator, 2-groups of order 2 10 with trivial Schur multiplier. The authors obtained seventeen deficiency zero presentations from eighteen groups in [4] by this method.…”

On Efficient Presentations for Infinite Sequences of 2-Groups with Fixed Coclass

“…Problem 5 in [8] asks whether there are infinitely many 3-generator p-groups with deficiency zero. A positive solution for this problem is given by Fouladi, Jamali & Orfi [6]. There are infinite coclass sequences of finite 2-groups with 3 generators and trivial Schur multiplicator.…”