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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

A factorization theorem for groups and Lie algebras


Author: Eugene Schenkman
Journal: Proc. Amer. Math. Soc. 68 (1978), 149-152
MSC: Primary 17B60; Secondary 20F05
MathSciNet review: 0469996
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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.


References [Enhancements On Off] (What's this?)

  • [1] Jens Mennicke, Einige endliche Gruppen mit drei Erzeugenden und drei Relationen, Arch. Math. (Basel) 10 (1959), 409–418 (German). MR 0113946
  • [2] Eugene Schenkman, Group theory, Robert E. Krieger Publishing Co., Huntington, N.Y., 1975. Corrected reprint of the 1965 edition. MR 0460422

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1978-0469996-4
Keywords: Generators, relations, factorization, abelian subgroup, abelian subalgebras, derived subgroup, derived subalgebra, nilpotent
Article copyright: © Copyright 1978 American Mathematical Society