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Deficiency zero presentations for certain perfect groups

Published online by Cambridge University Press:  14 November 2011

C. M. Campbell ,
E. F. Robertson ,
T. Kawamata ,
I. Miyamoto  and
P. D. Williams
C. M. Campbell
Affiliation:
Mathematical Institute, University of St Andrews, St Andrews KY16, 9SS, Scotland
E. F. Robertson
Affiliation:
Mathematical Institute, University of St Andrews, St Andrews KY16, 9SS, Scotland
T. Kawamata
Affiliation:
Faculty of Engineering, Yamanashi University, Takeda-4, Kofu, Japan
I. Miyamoto
Affiliation:
Faculty of Engineering, Yamanashi University, Takeda-4, Kofu, Japan
P. D. Williams
Affiliation:
Department of Mathematics, California State University, San Bernardino, CA 92407, U.S.A.
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Synopsis

In this paper we give two generator, two relation presentations for the following perfect groups which were not previously known to have deficiency zero: SL(2, 32), SL(2, 64), SL(2, 27), SL(2, 49), Â7, Ŝz(8), SL(2, 5) × SL(2, 5), SL(2, 5) × SL(2, 25). We also give two generator, two relation presentations for three other finite perfect groups, two having SL(2, 7) as an image and one having SL(2, 5) as an image. We also discuss presentations for certain other perfect groups which were known to have deficiency zero and find some neat new presentations for them.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 103 , Issue 1-2 , 1986 , pp. 63 - 71
Copyright
Copyright © Royal Society of Edinburgh 1986

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References

1Campbell, C. M. and Robertson, E. F.. A deficiency zero presentation for SL(2, p). Bull. London Math. Soc. 12 (1980), 17–20.CrossRefGoogle Scholar
2Campbell, C. M. and Robertson, E. F.. The efficiency of simple groups of order < 105. Comm. Algebra 10 (1982). 217–225.CrossRefGoogle Scholar
3Campbell, C. M. and Robertson, E. F.. On a class of groups related to SL(2,2n). In Computational Group Theory (ed. Atkinson, M. D.), pp. 43–49. (London: Academic Press, 1984).Google Scholar
4Campbell, C. M. and Robertson, E. F.. Presentations for simple groups G, 105 < ∣G∣ < 106. Comm. Algebra 12 (1984), 2643–2663.CrossRefGoogle Scholar
5Campbell, C. M. and Robertson, E. F.. A CAYLEY file of finite simple groups. EUROCAL '85, Lecture Notes in Computer Science 204, pp. 243–244 (Berlin: Springer-Verlag, 1985).Google Scholar
6Cannon, 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
7Johnson, D. L. and Robertson, E. F.. Finite groups of deficiency zero. In Homological Group Theory (ed. Wall, C. T. C.) LMS Lecture Notes 36, pp. 275–289 (Cambridge: Cambridge University Press, 1979).CrossRefGoogle Scholar
8Kenne, P. E.. Presentations for some direct products of groups. Bull. Austral. Math. Soc. 28 (1983), 131–133.CrossRefGoogle Scholar
9Kenne, P. E.. Comm. Algebra (to appear).Google Scholar
10McKay, J. and Young, K.-C.. The non-abelian simple groups G, ∣G∣ < 106-minimal generating pairs. Math. Comp. 33 (1979), 812–814.Google Scholar
11Robertson, E. F.. Efficiency of finite simple groups and their covering groups. Contemp. Math. 45 (1985), 287–294.CrossRefGoogle Scholar
12Sandlöbes, G.. Perfect groups of order less than 104. Comm. Algebra 9 (1981), 477–490.CrossRefGoogle Scholar
13Wiegold, J.. The Schur multiplier. In Groups-St Andrews 1981 (ed. Campbell, C. M. and Robertson, E. F.) LMS Lecture Notes 71, pp. 137–154 (Cambridge: Cambridge University Press, 1982).CrossRefGoogle Scholar
14Williams, P. D.. Ph.D. Thesis, University of St Andrews, 1982.Google Scholar