On stable torsion-free nilpotent groups
B. I. Zil'ber proved that any uncountably categorical torsion-free nilpotent nonabelian group is an algebraic group over an algebraically closed field. The authors show that any stable torsion-free nilpotent nonabelian group \(G\) interprets an infinite field. To prove this they proceed as follows. Using the descending chain condition for centralizers in \(G\) they find a definable subgroup \(H\) of \(G\) and \(a,b \in H\) such that (1) \(H\) is 2-step nilpotent, (2) \(C_ H(a)\) and \(C_ H(b)\) are abelian, (3) \(C_ H(a) \cap C_ H(b)= Z(H)\), (4) \([C_ H(a),b]=[a,C_ H(b)]=H'\); moreover, \((*)\) in \(H\) the centralizer of any noncentral element is abelian. A. I. Mal'tsev proved that under the conditions (1)-- (4) \(H\) interprets a ring \(R\) whose additive group is \(H'\). The authors observe that \((*)\) means exactly that the ring \(R\) is an integral domain. As a stable integral domain is a field, the result follows. The reviewer described the structure of groups \(H\) with \(H'=Z(H)\) satisfying (1)--(4); it turned out that they were very close to the groups of the form \(\mathbf{UT}_ 3 (R)\). The authors generalize the reviewer's method to describe the structure of arbitrary groups satisfying (1)--(4); as a corollary, they characterize stable groups \(H\) with torsion-free \(H'\) satisfying (1)--(4). It is also proved that if \(G\) is a stable torsion-free \(n\)-step nilpotent group, \(n \geq 2\), then \(G/Z_{n-1}(G)\) is divisible.
scientific article; zbMATH DE number 492855
scientific article