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On presentation of PSL (2, pn)

Part of: Linear algebraic groups and related topics Special aspects of infinite or finite groups Representation theory of groups

Published online by Cambridge University Press:  09 April 2009

C. M. Campbell ,
E. F. Robertson  and
P. D. Williams
C. M. Campbell
Affiliation:
University of St. AndrewsNorth Haugh St. Andrews Fife KY 16 9SS, Scotland
E. F. Robertson
Affiliation:
University of St. AndrewsNorth Haugh St. Andrews Fife KY 16 9SS, Scotland
P. D. Williams
Affiliation:
California State University5500 University Parkway San Bernardino, California 92407, U.S.A.
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Abstract

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We give presentations for the groups PSL(2, pn), p prime, which show that the deficiency of these groups is bounded below. In particular, for p = 2 where SL(2, 2n) = PSL(2, 2n), we show that these groups have deficiency greater than or equal to – 2. We give deficiency – 1 presentations for direct products of SL(2, 2n) for coprime ni. Certain new efficient presentations are given for certain cases of the groups considered.

MSC classification

Secondary: 20F05: Generators, relations, and presentations 20D06: Simple groups 20G40: Linear algebraic groups over finite fields
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 48 , Issue 2 , April 1990 , pp. 333 - 346
Copyright
Copyright © Australian Mathematical Society 1990

References

[1]Beetham, M. J., ‘A set of generators and relations for the groups PSL(2, q), q odd’, J. London Math. Soc. 3 (1971), 554–557.CrossRefGoogle Scholar
[2]Berlekamp, E. R., Algebraic coding theory (McGraw-Hill, 1968).Google Scholar
[3]Bussey, W. H., ‘Generational relations for the abstract group simply isomorphic with the group LF[2, pn]’, Proc. London Math. Soc. (2) 3 (1905), 296–315.CrossRefGoogle Scholar
[4]Campbell, C. M. and Robertson, E. F., ‘Classes of groups related to Fa, b, c’, Proc. Roy. Soc. Edinburgh Sect. A 78 (1978), 209–218.CrossRefGoogle Scholar
[5]Campbell, C. M. and Robertson, E. F., ‘A deficiency zero presentation for SL(2, p)’, Bull. London Math. Soc. 12 (1980), 17–20.CrossRefGoogle Scholar
[6]Campbell, C. M. and Robertson, E. F., ‘The efficiency of simple groups of order < 105’, Comm. Algebra 10 (1982), 217–225.CrossRefGoogle Scholar
[7]Campbell, C. M. and Robertson, E. F., ‘On a class of groups related to SL(2, 2n)’, Computational Group Theory, edited by Atkinson, M. D., pp. 43–49 (Academic Press, London, 1984).Google Scholar
[8]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
[9]Cohn, P. M., Algebra, Vol. 2 (Wiley, London, 1977).Google Scholar
[10]Huppert, B., Endliche Gruppen I (Springer-Verlag, Berlin, 1967).CrossRefGoogle Scholar
[11]Kenne, P. E., ‘Efficient presentations for three simple groups’, Comm. Algebra 14 (1986), 797–800.CrossRefGoogle Scholar
[12]Robertson, E. F., ‘Efficiency of finite simple groups and their covering groups’, Contemp. Math. 45 (1985), 287–294.CrossRefGoogle Scholar
[13]Robertson, E. F. and Williams, P. D., ‘Efficient presentations of the groups PSL(2, 2p) and SL(2, 2p)’, Bull. Canad. Math. Soc. 32 (1989), 3–10.CrossRefGoogle Scholar
[14]Sinkov, A., ‘A note on a paper by J. A. Todd’, Bull. Amer. Math. Soc. 45 (1939), 762–765.CrossRefGoogle Scholar
[15]Sunday, J. G., ‘Presentations of the groups SL(2, m) and PSL(2, m)’, Canad. J. Math. 24 (1972), 1129–1131.CrossRefGoogle Scholar
[16]Todd, J. A., ‘A second note on the linear fractional group’, J. London Math. Soc. 2 (1936), 103–107.CrossRefGoogle Scholar
[17]Williams, P. D., Presentations of linear groups (Ph. D. thesis, University of St. Andrews, 1982).Google Scholar
[18]Zassenhaus, H. J., ‘A presentation of the groups PSL(2, p) with three defining relations’, Canad. J. Math. 21 (1969), 310–311.CrossRefGoogle Scholar
[19]Zierler, N. and Brilihart, J., ‘On primitive trinomials (mod 2)’, Inform, and Control 13 (1968), 541–554.CrossRefGoogle Scholar