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