Archiv der Mathematik

, Volume 93, Issue 1, pp 23–28 | Cite as

On capability of finite abelian groups

  • Zoran Šunić


Baer characterized capable finite abelian groups (a group is capable if it is isomorphic to the group of inner automorphisms of some group) by a condition on the size of the factors in the invariant factor decomposition (the group must be noncyclic and the top two invariant factors must be equal). We provide a different characterization, given in terms of a condition on the lattice of subgroups. Namely, a finite abelian group G is capable if and only if there exists a family {H i } of subgroups of G with trivial intersection, such that the union generates G and all quotients G/H i have the same exponent. Other variations of this condition are also provided (for instance, the condition that the union generates G can be replaced by the condition that it is equal to G).

Mathematics Subject Classification (2000)

Primary 20K01 Secondary 20D30 


Capable groups Finite abelian groups 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Baer R.: Groups with preassigned central and central quotient group. Trans. Amer. Math. Soc. 44, 387–412 (1938)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bartholdi L., Šuniḱ Z.: On the word and period growth of some groups of tree automorphisms. Comm. Algebra 29, 4923–4964 (2001)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Brodie M.A., Chamberlain R.F., Kappe L.-C.: Finite coverings by normal subgroups. Proc. Amer. Math. Soc. 104, 669–674 (1988)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Grigorchuk R.I.: Degrees of growth of finitely generated groups and the theory of invariant means. Izv. Akad. Nauk SSSR Ser. Mat. 48, 938–985 (1984)MathSciNetGoogle Scholar
  5. 5.
    Z. Šuniḱ, On a class of periodic spinal groups of intermediate growth, PhD thesis, State University of New York at Binghamton, 2000.Google Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2009

Authors and Affiliations

  1. 1.Department of MathematicsTexas A&M UniversityCollege StationUSA

Personalised recommendations