Superstable groups; a partial answer to conjectures of Cherlin and Zil'ber (Q1080417)

The results of Cherlin and Cherlin/Shelah on groups of Morley rank/Shelah degree 2 and 3 are proved for superstable groups of U-rank \(\omega^{\alpha}2\) and \(\omega^{\alpha}3\). Use is made of the machinery developed in the paper reviewed above. The main results are: 1.) If G is connected, \(U(G)=\omega^{\alpha}2\) then G is solvable, and if G is not nilpotent then \(G/_{Z(G)}\) is isomorphic to the affine, connected centreless algebraic group of dimension 2 over an algebraically closed field F. 2.) If G is connected, \(U(G)=\omega^{\alpha}3\) then G is solvable or G is isomorphic to \(SL_ 2(F)\) or \(PSL_ 2(F)\) where F is an algebraically closed field. 3.) If G is connected, solvable and not nilpotent with \(U(G)=\omega^{\alpha}k\) then G interprets an algebraically closed field K with \(U(K)\geq \omega^{\alpha}\).