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

Abstract

Using computational methods, we first show that there are exactly eighteen 3-generator 2-groups of order 2¹⁰ with trivial Schur multiplier all having deficiency zero. We next generalize one of the groups obtained to exhibit two infinite classes of 3-generator, 3-relation finite 2-groups of high nilpotency class providing an affirmative answer to a problem posed by Havas et al.

Key Words:

• Deficiency zero
• Finite p-group
• Lower exponent-p central series
• Presentation
• Schur multiplier

2000 Mathematics Subject Classification:

• Primary 20F05, 20D15

ACKNOWLEDGMENT

The authors would like to thank Professor Bettina Eick for her helpful comments relating to the method described in Section 2 of the article. They also thank the referee for his useful comments and suggestions. The second author was partially supported by a grant from TMU (Iran).

Notes

Communicated by A. Turull.