Mathematical Notes

, Volume 105, Issue 1–2, pp 56–63 | Cite as

On a Family of Residually Finite Groups

  • D. I. MoldavanskiiEmail author


It is known that there exists a finitely generated residually finite group (for short, a residually F-group) the extension by which of some finite group is not a residually F-group. In the paper, it is shown that, nevertheless, every extension of a finite group by a finitely generated residually F-group is a Hopf group, and every extension of a center-free finite group by a finitely generated residually F-group is a residually F-group. If a finitely generated residually F-group G is such that every extension of an arbitrary finite group by G is a residually F-group, then a descending HNN-extension of the group G also has the same property, provided that it is a residually F-group.


residually finite groups HNN-extensions of groups 


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  1. 1.
    J. M. Corson and Th. J. Ratkovich, “A strong form of residual finiteness for groups,” J. Group Theory 9, 497–505 (2006).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    A. I. Mal’tsev, “On homomorphisms onto finite groups,” Uchen. Zap. Ivanov. Ped. Inst. 18, 49–60 (1958) [in Russian].Google Scholar
  3. 3.
    P. R. Hewitt, “Extensions of residually finite groups,” J. Algebra 163, 757–772 (1994).MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    B. Chandler and W. Magnus, The History of Combinatorial Group Theory. A Case Study in the History of Ideas (Springer–Verlag, New York, 1982; Mir,Moscow, 1985).CrossRefzbMATHGoogle Scholar
  5. 5.
    A. I. Mal’tsev, “On isomorphic matrix representations of infinite groups,” Mat. Sb. 8 (50) (3), 405–422 (1940) [Amer. Math. Soc. Transl. (2) 45, 1–18 (1965)].Google Scholar
  6. 6.
    S. Meskin, “Nonresidually finite one–relator groups,” Trans. Amer. Math. Soc. 164, 105–114 (1972).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    R. C. Lyndon and P. E. Schupp, Combinatorial Group Theory (Springer–Verlag, Berlin–New York, 1977; Mir,Moscow, 1980).zbMATHGoogle Scholar
  8. 8.
    D. I. Moldavanskii, “Residual finiteness of descending HNN–extensions of groups,” Ukr. Mat. Zh. 44, 842–845 (1992) [Ukr. Math. 44 (6), 758–760 (1992)].MathSciNetCrossRefGoogle Scholar

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© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.Ivanovo State UniversityIvanovoRussia

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