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