Groups with regular elementary 2-groups of automorphisms
Theorem 1. There exists a function \(f(x,y)\) of two natural variables such that any \(k\)-step solvable, periodic group \(G\) admitting a regular elementary group of automorphisms of order \(2^ n\) has an invariant series \(G=H_ 1\supseteq H_ 2\supseteq\cdots\supseteq H_{n+1}=1\), in which the factors are nilpotent and the nilpotent length of \(H_ i/H_{i+1}\) is at most \(f(i,k)\), \(1\leq i\leq n\). Theorem 2. Suppose \(G\) is a periodic group admitting a regular elementary group of automorphisms of order \(2^ n\). If some term of the derived series of \(G\) having a natural subscript is hypercentral, then \(G\) has an invariant series \(G=H_ 1\supseteq H_ 2\supseteq\dots\supseteq H_{n+1}=1\) in which all of the factors are hypercentral.
regular elementary groups of automorphisms
scientific article; zbMATH DE number 3974