Skip to main content Accessibility help
×
Home
Hostname: page-component-78dcdb465f-xl52z Total loading time: 0.286 Render date: 2021-04-15T15:20:46.439Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": false, "newCiteModal": false, "newCitedByModal": true }

On presentation of PSL (2, pn)

Published online by Cambridge University Press:  09 April 2009

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.
Rights & Permissions[Opens in a new window]

Abstract

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.

Type
Research Article
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), 554557.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), 296315.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), 209218.CrossRefGoogle Scholar
[5]Campbell, C. M. and Robertson, E. F., ‘A deficiency zero presentation for SL(2, p)’, Bull. London Math. Soc. 12 (1980), 1720.CrossRefGoogle Scholar
[6]Campbell, C. M. and Robertson, E. F., ‘The efficiency of simple groups of order < 105’, Comm. Algebra 10 (1982), 217225.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. 4349 (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), 6371.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), 797800.CrossRefGoogle Scholar
[12]Robertson, E. F., ‘Efficiency of finite simple groups and their covering groups’, Contemp. Math. 45 (1985), 287294.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), 310.CrossRefGoogle Scholar
[14]Sinkov, A., ‘A note on a paper by J. A. Todd’, Bull. Amer. Math. Soc. 45 (1939), 762765.CrossRefGoogle Scholar
[15]Sunday, J. G., ‘Presentations of the groups SL(2, m) and PSL(2, m)’, Canad. J. Math. 24 (1972), 11291131.CrossRefGoogle Scholar
[16]Todd, J. A., ‘A second note on the linear fractional group’, J. London Math. Soc. 2 (1936), 103107.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), 310311.CrossRefGoogle Scholar
[19]Zierler, N. and Brilihart, J., ‘On primitive trinomials (mod 2)’, Inform, and Control 13 (1968), 541554.CrossRefGoogle Scholar

Full text views

Full text views reflects PDF downloads, PDFs sent to Google Drive, Dropbox and Kindle and HTML full text views.

Total number of HTML views: 0
Total number of PDF views: 154 *
View data table for this chart

* Views captured on Cambridge Core between September 2016 - 15th April 2021. This data will be updated every 24 hours.

You have Access

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

On presentation of PSL (2, pn)
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

On presentation of PSL (2, pn)
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

On presentation of PSL (2, pn)
Available formats
×
×

Reply to: Submit a response


Your details


Conflicting interests

Do you have any conflicting interests? *