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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])
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Abstract

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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

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