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).
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The work presented here is partially supported by NSF/DMS-0805932.
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Šunić, Z. On capability of finite abelian groups. Arch. Math. 93, 23–28 (2009). https://doi.org/10.1007/s00013-009-0004-9
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DOI: https://doi.org/10.1007/s00013-009-0004-9