A factorization theorem for groups and Lie algebras

Abstract:

A proof is given for a generalization of a theorem of Mennicke that each member of a certain family of groups defined by generators and relations is finite. This leads to the following theorem on factorization of groups. Theorem. Let G be generated by abelian subgroups A, B, C, such that $[A,B] \leqslant A,[B,C] \leqslant B,[C,A] \leqslant C$; then the second derived group, $G''$ is nilpotent of class at most 3. Also proved is the analogue of the above theorem for Lie algebras.