Commutator conditions and splitting automorphisms for stable groups
There are various results in group theory of the following type: a group in a certain class is nilpotent provided it satisfies a certain commutator condition. The author proves a series of results of this type for stable groups -- an object that is extensively studied in model theoretic algebra. He shows that a stable soluble group satisfying some left normalized commutator condition \([x_{i_ 0},x_{i_ 1},\dots,x_{i_ n}]= 1\) with \(i_ 0\neq i_ 1\) is nilpotent. A stable group which is generically left (right) Engel is nilpotent. (A stable group is called generically left (right) Engel if its generic elements are left (right) Engel.) A stable soluble group with a generically splitting automorphism of prime order is nilpotent-by-finite. (An automorphism \(\varphi\) of order \(n\) of a stable group \(G\) is called generically splitting if \(xx^ \varphi\cdots x^{\varphi^{n-1}}=1\), for any generic element \(x\) of any type-definable characteristic subgroup of \(G\).) As a corollary, a stable soluble group with a generic element of prime order is nilpotent- by-finite.
scientific article; zbMATH DE number 438618