Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-04T20:08:41.619Z Has data issue: false hasContentIssue false
  • English
  • Français

On minimal irregular p-Groups

Published online by Cambridge University Press:  09 April 2009

J. R. J. Groves
J. R. J. Groves
Affiliation:
University of ManitobaWinnipeg, Manitoba, Canada
Rights & Permissions [Opens in a new window]

Extract

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

The study of minimal irregular p-groups was initiated by Mann [9]. He defines such a group to be a finite irregular p-group in which every proper section is regular. He then deduces a large number of properties of such groups and shows, by construction, that there is no bound on their exponent. In this note methods involving varieties of groups are used to show that all minimal irregular p-groups can be ‘built up’ from cyclic groups, groups of exponent p and a relatively small class of metabelian minimal irregular p-groups. To phrase this more precisely, say that a group P is derivable from a family of groups {Qi} if P is a homomorphic image of a subdirect product of the Qi. Then we will prove as Theorem 2.2: a minimal irregular p-group of exponent pn and nilpotency class c is derivable from a family of three groups: i) a cyclic group of order pn, ii) a 2-generator monolithic group of exponent p and class c, iii) a minimal irregular p-group whi h is metabelian, has class p and has exponent p2.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 16 , Issue 1 , August 1973 , pp. 78 - 89
Copyright
Copyright © Australian Mathematical Society 1973

References

[1]Brisley, W., ‘On varieties of metabelian p-groups and their laws’, J. Austral. Math. Soc. 7 (1967), 64–80.CrossRefGoogle Scholar
[2]Gupta, N. D., ‘Polynilpotent groups of prime exponent’, Bull. Amer. Math. Soc. 74 (1968), 559–561.CrossRefGoogle Scholar
[3]Higman, Graham, ‘Representations of general linear groups and varieties of groups’, Proc. Internat. Conf. Theory of Groups, Austral. Nat. Univ. Canberra, August 1965, 167–183, (Gordon and Breach, New York, 1967).Google Scholar
[4]Huppert, B., Endliche Gruppen I (Springer, New York, 1967).CrossRefGoogle Scholar
[5]Kostrikin, A. I., ‘Solution of the restricted Burnside problem for exponent 5’, Izv. Akad. Nauk SSSR. Ser. Mat. 19 (1955), 233–244 (Russian).Google Scholar
[6]Kostrikin, A. I., ‘The Burnside problem’, Izv. Akad. Nauk. SSSR Ser. Mat. 23 (1959), 3–34 (Russian) [English translation, Amer. Math. Soc. Transl. (2) 36 (1964), 63–99].Google Scholar
[7]Kovács, L. G. and Newman, M. F., ‘On critical groups’, J. Austral. Math. Soc. 6 (1966), 237–250.CrossRefGoogle Scholar
[8]Kovács, L. G. and Newman, M. F., ‘On non-Cross varieties of groups’, J. Austral. Math. Soc. 12 (1971), 129–144.CrossRefGoogle Scholar
[9]Mann, Avinoam, ‘Regular p-groups’, Israel J. Math. 10 (1971), 471–477.CrossRefGoogle Scholar
[10]Meier-Wunderli, H., ‘Metabelsche Gruppen’, Comment. Math. Helv. 25 (1951), 1–10.CrossRefGoogle Scholar
[11]Neumann, Hanna, Varieties of groups (Springer, Berlin, 1967).CrossRefGoogle Scholar
[12]Weichsel, Paul M., ‘Regular p-groups and varieties’, Math. Z. 95 (1967), 223–231.CrossRefGoogle Scholar
[13]Weichsel, Paul M., ‘Just irregular p-groups’, Israel J. Math. 10 (1971), 359–363.CrossRefGoogle Scholar