JournalsrsmupVol. 125pp. 81–105

On the Rarity of Quasinormal Subgroups

  • John Cossey

    Australian National University, Canberra, Australia
  • Stewart Stonehewer

    University of Warwick, Coventry, UK
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For each prime pp and positive integer nn, Berger and Gross have defined a finite pp-group G=HXG=HX, where HH is a core-free quasinormal subgroup of exponent pn1p^{n-1} and XX is a cyclic subgroup of order pnp^n. These groups are universal in the sense that any other finite pp-group, with a similar factorisation into subgroups with the same properties, embeds in GG. In our search for quasinormal subgroups of finite pp-groups, we have discovered that these groups GG have remarkably few of them. Indeed when pp is odd, those lying in HH can have exponent only pp, pn2p^{n-2} or pn1p^{n-1}. Those of exponent pp are nested and they all lie in each of those of exponent pn2p^{n-2} and pn1p^{n-1}.

Cite this article

John Cossey, Stewart Stonehewer, On the Rarity of Quasinormal Subgroups. Rend. Sem. Mat. Univ. Padova 125 (2011), pp. 81–105

DOI 10.4171/RSMUP/125-6