Hostname: page-component-848d4c4894-cjp7w Total loading time: 0 Render date: 2024-06-20T15:25:36.219Z Has data issue: false hasContentIssue false
  • English
  • Français

Recalcitrance in groups

Published online by Cambridge University Press:  17 April 2009

R.G. Burns ,
W.N. Herfort ,
S.-M. Kam ,
O. Macedońska  and
P.A. Zalesskii
R.G. Burns
Affiliation:
Department of Mathematics and Statistics, York University, Toronto, Canada M3J 1P3
W.N. Herfort
Affiliation:
Institut für Angew. Mathematik, Technische Universität, A-1040 Vienna, Austria
S.-M. Kam
Affiliation:
Kwi Shing Estate, New Territories, Hong Kong, China
O. Macedońska
Affiliation:
Institute of Mathematics, Silesian Technical University, 44-100 Gliwice, Poland
P.A. Zalesskii
Affiliation:
Department of Mathematics, University of Brasilia, 70910-900 Brasilia-DF, Brazil
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Motivated by a well-known conjecture of Andrews and Curtis, we consider the question as to how in a given n-generator group G, a given set of n “annihilators” of G, that is, with normal closure all of G, can be transformed by standard moves into a generating n-tuple. The recalcitrance of G is defined to be the least number of elementary standard moves (”elementary M-transformations”) by means of which every annihilating n-tuple of G can be transformed into a generating n-tuple. We show that in the classes of finite and soluble groups, having zero recalcitrance is equivalent to nilpotence, and that a large class of 2-generator soluble groups has recalcitrance at most 3. Some examples and remarks are included.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 60 , Issue 2 , October 1999 , pp. 245 - 251
Copyright
Copyright © Australian Mathematical Society 1999

References

[1]Andrews, J.J. and Curtis, M.L., ‘Extended Nielsen operations in free groups’, Amer. Math. Monthly 73 (1966), 21–28.CrossRefGoogle Scholar
[2]Burns, R.G. and Macedonska, O., ‘Balanced presentations of the trivial group’, Bull. London Math. Soc. 25 (1993), 513–526.CrossRefGoogle Scholar
[3]Miasnikov, A.D., ‘Genetic algorithms and the Andrews-Curtis conjecture’, (preprint).Google Scholar
[4]Robinson, D.J.S., A course in the theory of groups (Springer-Verlag, Berlin, Heidelberg, New York, 1982).Google Scholar