Skip to main content

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
Cancel
×
×
Home
  • Home
Article
  • Access

Groups of odd order in which every subnormal subgroup has defect at most two

  • John Cossey (a1)
Abstract

In 1980, McCaughan and Stonehewer showed that a finite soluble group in which every subnormal subgroup has defect at most two has derived length at most nine and Fitting length at most five, and gave an example of derived length five and Fitting length four. In 1984 Casolo showed that derived length five and Fitting length four are best possible bounds.

In this paper we show that for groups of odd order the bounds can be improved. A group of odd order with every subnormal subgroup of defect at most two has derived and Fitting length at most three, and these bounds are best possible.

    •
      • 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.

      Groups of odd order in which every subnormal subgroup has defect at most two
      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.

      Groups of odd order in which every subnormal subgroup has defect at most two
      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.

      Groups of odd order in which every subnormal subgroup has defect at most two
      Available formats
      ×
Copyright
COPYRIGHT: © Australian Mathematical Society 1991
References
Hide All
[1]Curtis, C. W. and Reiner, I., Representation Theory of Finite Groups and Associative Algebras, (Pure and Applied Mathematics 11, Interscience Publishers, New York, London, 1962).
[2]Casolo, C., ‘Gruppi finite risolubili in cui tutti i sottogrupi subnormali hanno difetto al pui 2’ Rend. Sem. Mat. Univ. Padova 71 (1984), 257–271.
[3]Gaschütz, W., ‘Grupprn in denen das Normalteilersein transitiv ist’, J. Reine Angew. Math., 198 (1957), 87–92.
[4]Gaschütz, W., ‘Endliche Gruppen mit treuen absolut-irreduziblen Darstellungen’, Math. Nachr. 12 (1954), 253–255.
[5]Heineken, H., ‘A class of three Engel groups’, J. Algebra 17 (1971), 341–345.
[6]Higman, G., ‘Complementation of abelian normal subgroups’, Publ. Math. Debrecen 4 (1955–1956), 455–458.
[7]Huppert, B., Endliche Gruppen I, (Die Grundlehren der Mathematischen Wissenschaften, Bd 134, Springer-Verlag, Berlin, Heidelberg, New York, 1967).
[8]Huppert, B. and Blackburn, N., Finite Groups II, (Die Grundlehren der Mathematischen Wissenschaften, Bd 242, Springer-Verlag, Berlin, Heidelberg, New York, 1982).
[9]Mahdavianary, S. K., ‘a special class of three Engel groups’, Arch. Math. 40 (1983), 193–199.
[10]McCaughan, D. J. and Stonehewer, S., ‘Finite soluble groups whose subnormal subgroups have defect at most two’, Arch. Math. 35 (1980), 56–60.
[11]Peng, T. A., ‘Finite groups with pronormal subgroups’, Amer. Math. Soc. 20 (1969), 232–234.
[12]Robinson, D. J. S., ‘A note on finite groups in which normality is transitive’, Proc. Amer. Math. Soc. 19 (1968), 933–937.
[13]Robinson, D. J. S., A Course in the Theory of Groups, (Graduate Texts in Mathematics 80, Springer-Verlag, New York, Heidelberg, Berlin, 1982).
[14]Taunt, D. R., ‘On A-groups’, Proc. Cambridge Phil. Soc. 45 (1949), 24–42.
[15]Zacher, G., ‘Caratterizzazione dei t-gruppi finiti risolubili’, Ricerche Mat. 1 (1952), 287–294.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of the Australian Mathematical Society
  • ISSN: 1446-7887
  • EISSN: 1446-8107
  • URL: /core/journals/journal-of-the-australian-mathematical-society
Please enter your name
Please enter a valid email address
Who would you like to send this to? *

CAPTCHA *

Skip to the audio challenge
×
MathJax

Keywords:

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 19 *
Loading metrics...

Abstract views

Total abstract views: 25 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 13th June 2018. This data will be updated every 24 hours.

Cancel
Confirm
×