The Upper and Lower Central Series in Linear Groups

A classical theorem of Reinhold Baer shows that any group which is finite over its k-Th centre has a finite (k + 1)-Th term of its lower central series. Although the converse statement is false in general, Philip Hall proved that for any group G the finiteness of γk + 1(G) implies that the index {2k}(G) is finite. Similar situations have been investigated when finiteness is replaced by suitable weaker conditions. Moreover, it was proved by YuriMerzljakov that Baer's theorem and its potential converse do hold for linear groups. The aim of this paper is to obtain results of the latter type for several other finiteness conditions. Finally, although a result of Baer type does not hold for the class of soluble-by-finite reduced minimax groups, we prove that for this class a theorem of Hall type is true in arbitrary groups.