Minimal nil-transformations of class two

On a metric minimal flow (X, a) which is a torus (K) extension of its largest almost periodic factor Z=X/K, the following conditions are equivalent. (i) (X, a) is a nil-transformation of the form (N/Γ, a) where K is central in N and [N, N]⊂K. (ii) E(X), the enveloping group of (X, a) is a nilpotent group of class 2. (iii) Any minimal subset Ω of X×X is invariant under the diagonal action of K and the quotient Ω/K=Z ₁, is the largest almost periodic factor of Ω. The enveloping groups of such flows are described and a corollary on cocycles of the circle into itself is deduced. Finally general minimal niltransformations of class two are shown to be of the form considered in condition (i) above (possibly with a different nilpotent group) and consequently we deduce that the class of minimal flows which are group factors of nil-transformations of class 2 is closed under factors.