Skip to main content Accessibility help
×
Home

Chains of Varieties

  • Narain Gupta (a1), Frank Levin (a2) and Akbar Rhemtulla (a3)

Summary

If is a variety of groups that can be denned by n-variable laws and (m) is the variety all of whose m-generator groups are in then there corresponds the chain: (1) (2) ≧ . . . ≧ (n) = . In this paper such chains are investigated to determine which of the inclusions are proper for certain varieties . In particular the inclusions are shown to be all proper for the varieties where is the variety of nilpotent-of-class-c groups, is the abelian variety and is the variety of centre-bymetabelian groups. For the inclusions are likewise proper but for the corresponding chain is:

    • 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.

      Chains of Varieties
      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.

      Chains of Varieties
      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.

      Chains of Varieties
      Available formats
      ×

Copyright

References

Hide All
1. Gruenberg, K. W., Residual properties of infinite soluble groups, Proc. London Math. Soc. 7 (1957), 2962.
2. Gupta, C. K., On 2-metabelian groups, Arch. Math. 19 (1968), 584587.
3. Gupta, C. K., Gupta, N. D., and Newman, M. F., Some finite nilpotent p-groups, J. Austral. Math. Soc. 9 (1969), 287288.
4. Gupta, N. D., Certain locally metanilpotent varieties of groups, Arch. Math. 22 (1969), 481484.
5. Heineken, H., Ueber ein Levisches Nilpotenzkriterium, Arch. Math. 12 (1961), 176178.
6. Kappe, W., Die A-Norm einer Gruppe, Illinois J. Math. 5 (1961), 187197.
7. Levi, F. W., Groups in which the commutator operation satisfies certain algebraic conditions, J. Indian Math. Soc. (N.S.) 6 (1942), 8797.
8. Levi, F. W. and Van, B. L. der Waerden, Ueber eine besondere Klasse von Gruppen, Abh. Math. Sem. Univ. Hamburg 9 (1932), 154158.
9. Levin, Frank, On some varieties of soluble groups, I. Math. Z. 85 (1964), 369372.
10. Macdonald, I. D., On certain varieties of groups, Math. Z. 76 (1961), 270282.
11. Macdonald, I. D., On certain varieties of groups, II. Math. Z. 78 (1962), 175188.
12. Macdonald, I. D. and Neumann, B. H., A Third-Engel h-group, J. Austral. Math. Soc. 7 (1967), 555569.
13. Magnus, W., Karass, A., and Solitar, D., Combinatorial group theory (Interscience, New York-London, 1966).
14. Neumann, B. H., On a conjecture of Hanna Neumann, Glasgow Math. J. 8 (1957), 1317.
15. Neumann, Hanna, Varieties of groups (Springer-Verlag, New York, 1967).
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

Chains of Varieties

  • Narain Gupta (a1), Frank Levin (a2) and Akbar Rhemtulla (a3)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed