Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-24T03:29:56.553Z Has data issue: false hasContentIssue false
  • English
  • Français

GENERALISING QUASINORMAL SUBGROUPS

Part of: Structure and classification of infinite or finite groups

Published online by Cambridge University Press:  23 February 2012

STEWART STONEHEWER
STEWART STONEHEWER*
Affiliation:
Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK (email: [email protected])
Rights & Permissions [Opens in a new window]

Abstract

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.

In Cossey and Stonehewer [‘On the rarity of quasinormal subgroups’, Rend. Semin. Mat. Univ. Padova125 (2011), 81–105] it is shown that for any odd prime p and integer n≥3, there is a finite p-group G of exponent pn containing a quasinormal subgroup H of exponent pn−1 such that the nontrivial quasinormal subgroups of G lying in H can have exponent only p, pn−1 or, when n≥4 , pn−2. Thus large sections of these groups are devoid of quasinormal subgroups. The authors ask in that paper if there is a nontrivial subgroup-theoretic property 𝔛 of finite p-groups such that (i) 𝔛 is invariant under subgroup lattice isomorphisms and (ii) every chain of 𝔛-subgroups of a finite p-group can be refined to a composition series of 𝔛-subgroups. Failing this, can such a chain always be refined to a series of 𝔛-subgroups in which the intervals between adjacent terms are restricted in some significant way? The present work embarks upon this quest.

Keywords

generalising quasinormality in finite p-groups

MSC classification

Secondary: 20E07: Subgroup theorems; subgroup growth
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 86 , Issue 1 , August 2012 , pp. 1 - 10
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2012

References

[1]Cossey, J. and Stonehewer, S. E., ‘On the rarity of quasinormal subgroups’, Rend. Semin. Mat. Univ. Padova 125 (2011), 81–105.CrossRefGoogle Scholar
[2]Cossey, J., Stonehewer, S. E. and Zacher, G., ‘Quasinormal subgroups of order p 2’, Ric. Mat. 57 (2008), 127–135.CrossRefGoogle Scholar
[3]Ore, O., ‘Structures and group theory I’, Duke Math. J. 3 (1937), 149–173.CrossRefGoogle Scholar
[4]Ore, O., ‘On the application of structure theory to groups’, Bull. Amer. Math. Soc. 44 (1938), 801–806.CrossRefGoogle Scholar
[5]Rose, J. S., A Course on Group Theory (Cambridge University Press, Cambridge, 1978).Google Scholar
[6]Schmidt, R., Subgroup Lattices of Groups (de Gruyter, Berlin, 1994).CrossRefGoogle Scholar
[7]Stern, M., Semimodular Lattices (Cambridge University Press, Cambridge, 1999).CrossRefGoogle Scholar