Skip to main content Accessibility help
×
Home
Hostname: page-component-544b6db54f-s4m2s Total loading time: 0.19 Render date: 2021-10-16T07:13:36.756Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": true, "newCiteModal": false, "newCitedByModal": true, "newEcommerce": true, "newUsageEvents": true }

Some subgroups of SL(3, z) generated by involutions

Published online by Cambridge University Press:  18 May 2009

Stephen P. Humphries
Affiliation:
Department of MathematicsBrigham Young UniversityProvoUtah 84602, U.S.A.
Rights & Permissions[Opens in a new window]

Extract

HTML view is not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

For R a commutative ring with identity 1 we let SL(n, R) denote the group of n × n integral matrices with determinant 1. A transvection T is an element of SL(n, R) which we represent (see [1]) as a pair (φ d) where φ ∈ (Rn)*, the dual space of Rn, d ∈ Rn, φ(d) = 0, and for all x ∈ Rn we have

T(x) = + φ(x) d.

Throughout this paper an involution is an element Y of SL(n, R) which has order two. Let n = 3 and R = Z and let C = diag(–1, –1, –1) be the central element of GL(3, Z).

Type
Research Article
Copyright
Copyright © Glasgow Mathematical Journal Trust 1990

References

1.Artin, E.Geometric algebra, (Interscience 1957).Google Scholar
2.Coxeter, H. S. M., Discrete groups generated by reflections, Ann. Math. 35 (1934) 588621.CrossRefGoogle Scholar
3.Brown, R. and Humphries, S. P., Orbits under symplectic transvections I and II, Proc. London Math. Soc. (3) 52 (1986), 517531and 532–556.CrossRefGoogle Scholar
4.Humphries, S. P., Free subgroups of SL(n, Z), n > 2, generated by transvections, J. Algebra 116 (1988), 155162.CrossRefGoogle Scholar
5.Newman, M., Integral matrices (Academic Press 1972).Google Scholar
You have Access
3
Cited by

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.

Some subgroups of SL(3, z) generated by involutions
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.

Some subgroups of SL(3, z) generated by involutions
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.

Some subgroups of SL(3, z) generated by involutions
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *