Skip to main content Accessibility help
×
Home
Hostname: page-component-5bf98f6d76-rtbc9 Total loading time: 0.311 Render date: 2021-04-20T18:05:00.075Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": false, "newCiteModal": false, "newCitedByModal": true }

Enumerating p-Groups

Published online by Cambridge University Press:  09 April 2009

Bettina Eick
Affiliation:
Fachbereich Mathematik Universität Kassel Heinrich-Plett-Str. 40 34132 Kassel Germany e-mail: eick@mathematik.uni-kassel.de
E. A. O'Brien
Affiliation:
Department of Mathmatics University of Auckland Private Bag 92019 Auckland New Zealand e-mail: obrien@math.auckland.ac.nz
Rights & Permissions[Opens in a new window]

Abstract

We present a new algorithm which uses a cohomological approach to determine the groups of order pn, where p is a prime. We develop two methods to enumerate p-groups using the Cauchy-Frobenius Lemma. As an application we show that there are 10 494213 groups of order 29.

1991 Mathematics subject classification (Amer. Math. Soc.): primary 20D15.

MSC classification

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1999

References

[1]Besche, H. U. and Eick, B., ‘Construction of finite groups’, J. Symbolic Comput. 27 (1999), 387404.CrossRefGoogle Scholar
[2]Besche, H. U. and Eick, B., ‘The groups of order at most 1000 except 512 and 768’, J. Symbolic Comput. 27 (1999), 405413.CrossRefGoogle Scholar
[3]Besche, H. U. and Eick, B., ‘The groups of order q n p’, Technical Report.Google Scholar
[4]Birkhoff, G., ‘Subgroups of Abelian groups’, Proc. London Math. Soc. 38 (1934), 385401.Google Scholar
[5]Bosma, W., Cannon, J. and Playoust, C., ‘The MAGMA algebra system I: The user language’, J. Symbolic Comput. 24 (1997), 235265.CrossRefGoogle Scholar
[6]Higman, G., ‘Enumerating p-groups I: Inequalities’, Proc. London Math. Soc. 10 (1960), 2430.CrossRefGoogle Scholar
[7]James, R., Newman, M. F. and O'Brien, E. A., ‘The groups of order 128’, J. Algebra 129 (1990), 136158.CrossRefGoogle Scholar
[8]Neubüser, J., ‘Investigations of groups on computers’, in: Computational problems in abstract algebra (Pergamon Press, Oxford, 1967) pp. 119.Google Scholar
[9]Newman, M. F., ‘Determination of groups of prime-power order’, in: Group theory (Canberra 1975), Lecture Notes in Math. 573 (Springer, Berlin, 1977) pp. 7384.CrossRefGoogle Scholar
[10]O'Brien, E. A., ‘The p-group generation algorithm’, J. Symbolic Comput. 9 (1990), 677698.CrossRefGoogle Scholar
[11]O'Brien, E. A., ‘The groups of order 256’, J. Algebra 143 (1991), 219235.CrossRefGoogle Scholar
[12]O'Brien, E. A., ‘Compouting automorphism groups of p-groups’, in: Computational algebra and number theory (Sydney, 1992) (Kluwer Academic Publ., Dordrecht, 1995) pp. 8390.CrossRefGoogle Scholar
[13]O'Brien, E. A., ‘Bibliography on the determination of finite groups’, Available from http://www.math.auckland.ac.nz/~obrien.Google Scholar
[14]Robinson, D. J., A course in the theory of groups, Graduate Texts in Math. 80, 2nd Edition (Springer, New York, 1996).CrossRefGoogle Scholar
[15]Sims, C. C., ‘Enumerationg p-groups’, Proc. London Math. Soc. 15 (1965), 151166.CrossRefGoogle Scholar
[16]Sims, C. C., Computation with finite finitely presented groups (Cambridge University Press, New York, 1994).CrossRefGoogle Scholar
[17]The GAP Team, GAP - Groups, algorithms, and programming, version 4, Lehrstuhl D für Mathematik, RWTH Aachen, and School of Mathematical and Computational Sciences, University of St Andrews, 1999.Google Scholar
[18]Vaughan-Lee, M. R., ‘An aspect of the nilpotent quotient algorithm’, in: Computational group theory (Durham, 1982) (Academic Press, London, 1984) pp. 7683.Google Scholar
[19]Wegner, A., The construction of finite soluble factor groups of finitely presented groups and its applications (Ph.D. Thesis, University of St Andrews, 1992).Google 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: 282 *
View data table for this chart

* Views captured on Cambridge Core between September 2016 - 20th 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.

Enumerating p-Groups
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.

Enumerating p-Groups
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.

Enumerating p-Groups
Available formats
×
×

Reply to: Submit a response


Your details


Conflicting interests

Do you have any conflicting interests? *