Hostname: page-component-cb9f654ff-5kfdg Total loading time: 0 Render date: 2025-08-22T12:10:29.400Z Has data issue: false hasContentIssue false
  • English
  • Français

The efficiency of PSL(2, p)3 and other direct products of groups

Published online by Cambridge University Press:  18 May 2009

C. M. Campbell ,
I. Miyamoto ,
E. F. Robertson  and
P. D. Williams
C. M. Campbell
Affiliation:
University of St. Andrews, North Haugh, St. Andrews, Fife KY16 9SS, Scotland
I. Miyamoto
Affiliation:
Faculty of Engineering, Yamanashi University, Takeda-4, Kofu, Japan
E. F. Robertson
Affiliation:
University of St. AndrewsNorth Haugh, St. Andrews, Fife KY16 9SS, Scotland
P. D. Williams
Affiliation:
Department of Mathematics, California State University, San BernardinoCalifornia 92407, U.S.A.
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.

A finite group G is efficient if it has a presentation on n generators and n + m relations, where m is the minimal number of generators of the Schur multiplier M (G)of G. The deficiency of a presentation of G is r–n, where r is the number of relations and n the number of generators. The deficiency of G, def G, is the minimum deficiency over all finite presentations of G. Thus a group is efficient if def G = m. Both the problem of efficiency and the converse problem of inefficiency have received considerable attention recently; see for example [1], [3], [14] and [15].

Information

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 39 , Issue 3 , September 1997 , pp. 259 - 268
Copyright
Copyright © Glasgow Mathematical Journal Trust 1997

References

REFERENCES

1.Baik, Y. G. and Pride, S. J., On the efficiency of Coxeter groups, Bull. London Math. Soc. 29 (1997), 32–36.CrossRefGoogle Scholar
2.Bogley, W. A. and Pride, S. J., Computing generators of π2, in Two-dimensional Homotopy and Combinatorial Group theory (Hog-Angeloni, C., Metzler, W. and Sieradski, A. J. eds.), London Math. Soc. Lecture Notes 197 (Cambridge University Press, Cambridge, 1995), 157–188.Google Scholar
3.Brookes, M. J., Campbell, C. M. and Robertson, E. F., Efficiency and direct products of groups, in Proc. Groups - Korea 1994 (Kim, A. C. and Johnson, D. L. eds.), (Walter de Gruyter, 1995) 25–33.Google Scholar
4.Campbell, C. M., Kawamata, T., Miyamoto, I., Robertson, E. F. and Williams, P. D., Deficiency zero presentations for certain perfect groups, Proc. Roy. Soc. Edinburgh Sect. A 103 (1986), 63–71.CrossRefGoogle Scholar
5.Campbell, C. M., Robertson, E. F. and Williams, P. D., On the efficiency of some direct powers of groups, in Groups-Canberra 1989 (Kovács, L. G. ed.), Lecture Notes in Mathematics No. 1456 (Springer-Verlag, Berlin, 1990), 106–113.CrossRefGoogle ScholarPubMed
6.Campbell, C. M., Robertson, E. F. and Williams, P. D., Efficient presentations of the groups PSL(2, p) × PSL(2, p), p prime, J. London Math. Soc. (2) 41 (1990), 69–77.CrossRefGoogle Scholar
7.Campbell, C. M., Robertson, E. F. and Williams, P. D., Efficient presentations of direct powers of imperfect groups, Algebra Colloq. 4 (1997), 21–27.Google Scholar
8.Cannon, J. J., McKay, J. and Young, K. C., The non-abelian simple groups G, |G| < 105—presentations, Comm. Algebra 7 (1979), 1397–1406.CrossRefGoogle Scholar
9.Conway, J. H., Coxeter, H. S. M. and Shephard, G. C., The centre of a finitely generated group, Tensor 25 (1972), 405–418.Google Scholar
10.Hert, T. and Williams, P. D., A note on a presentation of PGL (2,p) to appear.Google Scholar
11.Jamali, A., Computing with simple groups: maximal subgroups and presentations, Ph.D. thesis, University of St Andrews (1988).Google Scholar
12.Kenne, P. E., Presentations for some direct products of groups, Bull. Austral. Math. Soc. 28 (1983), 131–133.CrossRefGoogle Scholar
13.Kenne, P. E., Minimal group presentations: a computational approach, Ph.D. Thesis, Australian National University, 1991.Google Scholar
14.Kovács, L. G., Finite groups with trivial multiplicator and large deficiency, in Proc. Groups – Korea 1994 (Kim, A. C. and Johnson, D. L. eds.), (Walter de Gruyter, 1995), 211–225.Google Scholar
15.Robertson, E. F., Thomas, R. M. and Wotherspoon, C. I., A class of inefficient groups with symmetric presentations, in Proc. Groups – Korea 1994 (Kim, A. C. and Johnson, D. L. eds.), (Walter de Gruyter, 1995), 277–284.Google Scholar
16.Sunday, J. G., Presentations of the groups SL(2, m) and PSL(2, m), Canad. J. Math 24 (1972), 1129–1131.CrossRefGoogle Scholar
17.Suzuki, M., Group theory I, Grundlehren der Mathematischen Wissenschaften 247 (Springer-Verlag, 1982).CrossRefGoogle Scholar
18.Wiegold, J., The multiplicator of a direct product, Quart. J. Oxford (2) 22 (1971), 103–105.CrossRefGoogle Scholar
19.Wiegold, J., The Schur multiplier, in Groups – St Andrews 1981, (Campbell, C. M. and Robertson, E. F. eds.), London Mathematical Society Lecture Notes 71 (Cambridge University Press, Cambridge, 1982), 137–154.CrossRefGoogle Scholar