the probability that the commutator of two group elements is equal to a given element has been introduced in literature few years ago. several authors have investigated this notion with methods of the representation theory and with combinatorial techniques. here we illustrate that a wider context may be considered and show some structural restrictions on the group.

A degree is saturated bounding if it can compute a saturated model for any complete, decidable theory whose types are all computable. All high and PA degrees are saturated bounding (following from results of Jockusch and MacIntyre-Marker.) We show that for every n, no lown c.e. degree is saturated bounding, extending the previous known result that 0 is not saturated bounding, by Millar.

Important examples of Π 1 classes of functions f ∈ ω are the classes of sets (elements of 2) which separate a given pair of disjoint r.e. sets: S2(A0, A1) := { f ∈ 2 : (∀i < 2)(∀x ∈ Ai)f(x) 6= i }. A wider class consists of the classes of functions f ∈ k which in a generalized sense separate a k-tuple of r.e. sets (not necessarily pairwise disjoint) for each k ∈ ω: Sk(A0, . . . , Ak−1) := { f ∈...

A nontrivial automorphism of the Turing degrees is constructed.