Abstract
We prove that every free nonabelian group has a finitely generated isolated subgroup not separable in the class of nilpotent groups. This enables us to give a negative answer to the following question by D. I. Moldavanskii in the “Kourovka Notebook”: Is it true that every finitely generated \(p\prime \)-isolated subgroup of a free group is separable in the class of finite p-groups?
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Mal'tsev A. I., “On homomorphisms onto finite groups,” Uchen. Zap. Ivanovsk. Ped. Inst., 18,No. 5 (1958), 49-60 (also in “Selected Papers,” Vol. 1, Algebra, 1976, 450–462).
Hall M., Jr., “Coset representations in free groups,” Trans. Amer. Math. Soc., 67,No. 2, 421-432 (1949).
The Kourovka Notebook (Unsolved Problems in Group Theory), 15th ed., Inst. Math. SO RAN, Novosibirsk (2002).
Loginova E. D., “Residual finiteness of the free product of two groups with commuting subgroups,” Sibirsk. Mat. Zh., 40,No. 2, 395-407 (1999).
Kargapolov M. I. and Merzlyakov Yu. I., Fundamentals of the Theory of Groups, Springer-Verlag, New York (1979).
Neumann H., Varieties of Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 37, Springer-Verlag, Berlin; Heidelberg; New York (1967).
Rights and permissions
About this article
Cite this article
Bardakov, V.G. On D. I. Moldavanskii's Question About p-Separable Subgroups of a Free Group. Siberian Mathematical Journal 45, 416–419 (2004). https://doi.org/10.1023/B:SIMJ.0000028606.51473.f7
Issue Date:
DOI: https://doi.org/10.1023/B:SIMJ.0000028606.51473.f7