Finiteness conditions for a group of finite exponent
A periodic group \(G\) all of whose \(d\)-generator subgroups (for fixed \(d > 1\)) are solvable need not itself be solvable (Golod), and similarly for nilpotence. However if \(G\) is finite, then the solvability of every 2- generator subgroup entails the solvability of \(G\) (Thompson), and likewise for nilpotence. By strengthening the hypothesis the author obtains in the present paper similar results without the finiteness assumption. Let \(d\), \(d'\) (\(d \geq 1\), \(d' \geq 0\)) be integers. A group is called \((d,d')\)-solvable if each \(d\)-generator subgroup is solvable of length \(\leq d'\). Theorem 1. A periodic \((d,d-1)\)-solvable group (for some \(d \geq 1\)) is locally finite (and so certainly locally solvable). The analogous result for a \((d,2^ d-2)\)-nilpotent group (for some \(d \geq 1\)) also holds. An application is given to the question of \textit{Hanna Neumann} [from her book ``Varieties of Groups'' (1967; Zbl 0251.20001)] as to whether for each integer \(e > 0\) there exists an integer \(d > 0\) such that for a group of exponent \(e\) to be locally finite it suffices that the \(d\)- generator subgroups be finite. It is shown that for \(e \leq 6\) or \(e = 10\), 12, 15 or 20, the answer is in the affirmative. Finally a result on \((d,2^ d-2)\)-nilpotent groups of finite exponent is proved using the result of Zelmanov on the nilpotence of Lie algebras over a field of characteristic zero satisfying an Engel condition.
\((d,2^ d-2)\)-nilpotent groups of finite exponent
scientific article; zbMATH DE number 169078