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GENERALISING QUASINORMAL SUBGROUPS

  • STEWART STONEHEWER (a1)

Abstract

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.

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Copyright

References

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[1]Cossey, J. and Stonehewer, S. E., ‘On the rarity of quasinormal subgroups’, Rend. Semin. Mat. Univ. Padova 125 (2011), 81105.
[2]Cossey, J., Stonehewer, S. E. and Zacher, G., ‘Quasinormal subgroups of order p 2’, Ric. Mat. 57 (2008), 127135.
[3]Ore, O., ‘Structures and group theory I’, Duke Math. J. 3 (1937), 149173.
[4]Ore, O., ‘On the application of structure theory to groups’, Bull. Amer. Math. Soc. 44 (1938), 801806.
[5]Rose, J. S., A Course on Group Theory (Cambridge University Press, Cambridge, 1978).
[6]Schmidt, R., Subgroup Lattices of Groups (de Gruyter, Berlin, 1994).
[7]Stern, M., Semimodular Lattices (Cambridge University Press, Cambridge, 1999).
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Keywords

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GENERALISING QUASINORMAL SUBGROUPS

  • STEWART STONEHEWER (a1)

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