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Archiv der Mathematik

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

On capability of finite abelian groups

  • Zoran Šunić
Article

Abstract

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 

Keywords

Capable groups Finite abelian groups 

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Copyright information

© Birkhäuser Verlag Basel/Switzerland 2009

Authors and Affiliations

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

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