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Finite Groups With Few Relative Tensor or Exterior Degrees

  • Francesco G. RussoEmail author
  • Peyman Niroomand
Article
  • 11 Downloads

Abstract

A peculiar structure is present in a finite group G, when \(\mathcal {D}(G)=\{d(H,G) \ | \ H \ \text{ is } \text{ a } \text{ subgroup } \text{ of } \ G\}\) is small enough (here d(HG) denotes the relative commutativity degree). Recent contributions show that G has elementary abelian quotients, when \(|\mathcal {D}(G)| \le 4\). We introduce a similar problem for the relative exterior degree \(d^\wedge (H,G)\) and for the relative tensor degree \(d^\otimes (H,G)\). Theorems of structure are shown when G has a small number of relative tensor (or exterior) degrees. Among other things, we give new estimations for the gap \(d^\wedge (H,G)-d^\otimes (H,G)\) and for the arithmetic average \((d^\wedge (H,G)+d^\otimes (H,G))/2\).

Keywords

Relative tensor degree Commutativity degree Exterior degree 

Mathematics Subject Classification

Primary: 20J99 20D15 Secondary: 20D60 20C25 

Notes

Acknowledgements

We thank the referee for some useful comments, which help to clarify the exposition of the material. The first author thanks NRF for the Grant No. 118517 and both NRF and MAECI for the Grant No. 113144.

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Copyright information

© Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2019

Authors and Affiliations

  1. 1.Department of Mathematics and Applied MathematicsUniversity of Cape TownCape TownSouth Africa
  2. 2.School of Mathematics and Computer ScienceDamghan UniversityDamghanIran

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