Hostname: page-component-848d4c4894-x5gtn Total loading time: 0 Render date: 2024-06-01T16:23:31.165Z Has data issue: false hasContentIssue false
  • English
  • Français

Finite one-relator products of two cyclic groups with the relator of arbitrary length

Part of: Special aspects of infinite or finite groups

Published online by Cambridge University Press:  09 April 2009

C. M. Campbell ,
P. M. Heggie ,
E. F. Robertson  and
R. M. Thomas
C. M. Campbell
Affiliation:
Mathematical Institute University of St AndrewsSt Andrews KY16 9SS, Scotland
P. M. Heggie
Affiliation:
Mathematical Institute University of St AndrewsSt Andrews KY16 9SS, Scotland
E. F. Robertson
Affiliation:
Mathematical Institute University of St AndrewsSt Andrews KY16 9SS, Scotland
R. M. Thomas
Affiliation:
Department of Computing Studies University of LeicesterLeicester LE1 7RH, England
Rights & Permissions [Opens in a new window]

Abstract

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.

In this paper we consider the groups G = G(α, n) defined by the presentations . We derive a formula for [G′: ″] and determine the order of G whenever n ≦ 7. We show that G is a finite soluble group if n is odd, but that G can be infinite when n is even, n ≧ 8. We also show that G(6, 10) is a finite insoluble group involving PSU(3, 4), and that the group H with presentation is a finite group of deficiency zero of order at least 114,967,210,176,000.

MSC classification

Secondary: 20F05: Generators, relations, and presentations
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 53 , Issue 3 , December 1992 , pp. 352 - 368
Copyright
Copyright © Australian Mathematical Society 1992

References

[1]Aschbacher, M. and Guralnick, R., ‘Some applications of the first cohomology group’, J. Algebra 90 (1984), 446–460.CrossRefGoogle Scholar
[2]Baumslag, G., Morgan, J. W. and Shalen, P. B., ‘Generalized triangle groups’, Math. Proc. Camb. Phil. Soc. 102 (1987), 25–31.CrossRefGoogle Scholar
[3]Brunner, A. M., ‘The determination of Fibonacci groups’, Bull. Austral. Math. Soc. 11 (1974), 11–14.CrossRefGoogle Scholar
[4]Campbell, C. M., Coxeter, H. S. M. and Robertson, E. F., ‘Some families of finite groups having two generators and two relations’, Proc. Roy. Soc. London 357A (1977), 423–438.Google Scholar
[5]Campbell, C. M., Doostie, H. and Robertson, E. F., ‘Fibonacci length of generating pairs in groups’, in: Applications of Fibonacci numbers (eds. Bergum, G. E., Philippou, A. N. and Horadam, A. F.) (Kluwer, 1990), pp. 27–35.CrossRefGoogle Scholar
[6]Campbell, C. M., Heggie, P. M., Robertson, E. F. and Thomas, R. M., One-relator products of cyclic groups and Fibonacci-like sequences (Technical Report 35, Department of Computing Studies, University of Leicester, 04 1990).Google Scholar
[7]Campbell, C. M., Heggie, P. M., Robertson, E. F. and Thomas, R. M., ‘One-relator products of cyclic groups and Fibonacci-like sequences’, in: Applications of Fibonacci numbers (eds. Bergum, G. E., Phillippou, A. N. and Horadam, A. F.) (Kulwer, 1991), pp. 63–68.CrossRefGoogle Scholar
[8]Campbell, C. M. and Robertson, E. F., ‘Classes of groups related to Fa, b, c’, Proc. Roy. Soc. Edinburgh 78A (1978), 209–218.CrossRefGoogle Scholar
[9]Campbell, C. M. and Robertson, E. F., ‘On 2-generator 2-relation soluble groups’, Proc. Edinburgh Math. Soc. 23 (1980), 269–273.CrossRefGoogle Scholar
[10]Campbell, C. M. and Robertson, E. F., ‘Groups related to Fa, b, c involving Fibonacci numbers’, in: The geometric vein (eds. Davis, C., Grünbaum, B. and Sherk, F. A.) (Springer-Verlag, 1982), pp. 569–576.Google Scholar
[11]Campbell, C. M. and Robertson, E. F., ‘On the Fa, b, c conjecture’, Mitt. Math. Sem. Giessen 164 (1984), 25–36.Google Scholar
[12]Campbell, C. M., Robertson, E. F. and Thomas, R. M., ‘Fibonacci numbers; and groups’, in: Applications of Fibonacci numbers (eds. Horadam, A. F., Philippou, A. N. and Bergum, G. E.) (D. Reidel, 1987), pp. 45–59.Google Scholar
[13]Campbell, C. M., Robertson, E. F. and Thomas, R. M., ‘On groups related to Fibonacci groups’, in: Group theory (eds. Cheng, K. N. and Leong, Y. K.) (Walter de Gruyter & Co., 1989), pp. 323–331.CrossRefGoogle Scholar
[14]Campbell, C. M. and Thomas, R. M., ‘On (2, n)-groups related to Fibonacci groups’, Israel J. Math. 58 (1987), 370–380.CrossRefGoogle Scholar
[15]Cannon, J. J., ‘An introduction to the group theory language Cayley’, in: Computational group theory (ed. Atkinson, M. D.) (Academic Press, 1984), pp. 145–183.Google Scholar
[16]Chalk, C. P. and Johnson, D. L., ‘The Fibonacci groups II’, Proc. Roy. Soc. Edinburgh 77A (1977), 79–86.CrossRefGoogle Scholar
[17]Conway, J. H. et al. , ‘Solution to advanced problem 5327’, Amer. Math. Monthly 74 (1967), 91–93.CrossRefGoogle Scholar
[18]Doostie, H., Fibonacci-type sequences and classes of groups (Ph. D. Thesis, University of StAndrews, 1988).Google Scholar
[19]Fine, B., Howie, J. and Rosenberger, G., ‘One relator quotients and free products of cyclics’, Proc. Amer. Math. Soc. 102 (1988), 249–254.CrossRefGoogle Scholar
[20]Havas, G., ‘A Reidemeister-Schreier program’, in: Proc. Second Intern. Conf. Theory of Groups, Canberra 1973 (ed. Newman, M. F.) (Lecture Notes in Mathematics 372, Springer-Verlag, 1974), pp. 347–356.CrossRefGoogle Scholar
[21]Havas, G., ‘Computer aided determination of a Fibonacci group’, Bull. Austral. Math. Soc. 15 (1976), 297–305.CrossRefGoogle Scholar
[22]Havas, G., Kenne, P. E., Richardson, J. S. and Robertson, E. F., ‘A Tietze transformation program’, in: Computational group theory (ed. Atkinson, M. D.) (Academic Press, 1984), pp. 69–73.Google Scholar
[23]Helling, H., Kim, A. C. and Mennicke, J. L., ‘On Fibonacci groups’, preprint.Google Scholar
[24]Holt, D. F. and Plesken, W., Perfect groups (Oxford University Press, 1989).Google Scholar
[25]Johnson, D. L. and Robertson, E. F., ‘Finite groups of deficiency zero’, in: Homological Group Theory (ed. Wall, C. T. C.) (London Math. Soc. Lecture Note Series 36, Cambridge University Press, 1979) pp. 275–289.CrossRefGoogle Scholar
[26]Johnson, D. L., Wamsley, J. W. and Wright, D., ‘The Fibonacci groups’, Proc. London Math. Soc. 29 (1974), 577–592.CrossRefGoogle Scholar
[27]Lyndon, R. C., ‘On a family of infinite groups introduced by Conway’, unpublished.Google Scholar
[28]McKay, J. and Young, K-C., ‘The nonabelian simple groups G, |G| < 106-minimal generating pairs’, Math. Comp. 33 (1979), 812–814.Google Scholar
[29]Newman, M. F., ‘Proving a group infinite’, Archiv. Math. 54 (1990), 209–211.CrossRefGoogle Scholar
[30]Newman, M. F. and O'Brien, E. A., ‘A computer-aided analysis of some finitely-presented groups’, J. Austral. Math. Soc.Google Scholar
[31]Perkel, M., ‘Groups of type Fa, b, -c’, Israel J. Math. 52 (1985), 167–176.CrossRefGoogle Scholar
[32]Pride, S. J., ‘Groups with presentations in which each defining relator involves exactly two generators’, J. London Math. Soc. 36 (1987), 245–256.CrossRefGoogle Scholar
[33]Steinberg, R., ‘Generators for simple groups’, Canad. J. Math. 14 (1962), 277–283.CrossRefGoogle Scholar
[34]Thomas, R. M., ‘The Fibonacci groups F(2, 2m)’, Bull. London Math. Soc. 21 (1989), 463–465.CrossRefGoogle Scholar
[35]Thomas, R. M., ‘The Fibonacci groups revisited’, in: Proceedings of Groups-St. Andrews 1989, vol. 2, (eds. Campbell, C. M. and Robertson, E. F.) (London Math. Soc. Lecture Note Series 160, Cambridge University Press, 1991) pp. 445–454.CrossRefGoogle Scholar
[36]Wilcox, H. J., ‘Fibonacci sequences of period n in groups’, Fibonacci Quart. 24 (1986), 356–361.Google Scholar