Finite Groups With Few Relative Tensor or Exterior Degrees

A peculiar structure is present in a finite group G , when D ( G ) = { d ( H , G ) | H is a subgroup of G } is small enough (here d ( H , G ) denotes the relative commutativity degree). Recent contributions show that G has elementary abelian quotients, when | D ( G ) | ≤ 4 . We introduce a similar problem for the relative exterior degree d ∧ ( H , G ) and for the relative tensor degree d ⊗ ( 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 ∧ ( H , G ) - d ⊗ ( H , G ) and for the arithmetic average ( d ∧ ( H , G ) + d ⊗ ( H , G ) ) / 2 .