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Finite Groups With Few Relative Tensor or Exterior Degrees

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 \((d^\wedge (H,G)+d^\otimes (H,G))/2\).

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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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Correspondence to Francesco G. Russo.

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Communicated by V. Ravichandran.

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Russo, F.G., Niroomand, P. Finite Groups With Few Relative Tensor or Exterior Degrees. Bull. Malays. Math. Sci. Soc. 43, 3201–3219 (2020). https://doi.org/10.1007/s40840-019-00861-2

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Keywords

  • Relative tensor degree
  • Commutativity degree
  • Exterior degree

Mathematics Subject Classification

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