On Capability of Finite Abelian Groups

Baer characterized capable finite abelian groups (a group is capable if it is isomorphic to the quotient of some group by its center) 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 {Hi} of subgroups of G with trivial intersection, such that the union generates G and all the quotients G/Hi have the same exponent. The condition that the family of subgroups generates G may be replaced by the condition that the family covers G and the condition that the quotients have the same exponent may be replaced by the condition that the quotients are isomorphic. A class of finite groups with a certain property played a crucial role in the construction of a large family of finitely generated torsion groups of intermediate growth [BŠ01] that generalize the examples of Grigorchuk [Gri84]. Namely, Bartholdi and the author used finite groups B that satisfy the following condition. There exists a family of normal subgroups {Ni}i∈I of B such that (i) ⋂ i∈I Ni = 1, (ii) ⋃ i∈I Ni = B, (iii) B/Ni ∼= B/Nj, for i, j ∈ I. In his dissertation [Šun00] the author conjectured that the class of finite abelian groups that have this property is precisely the class of noncyclic abelian groups for which the top two factors in the invariant factor decomposition are equal. We prove this conjecture here. By an earlier result of Baer [Bae38], it follows that this class of finite abelian groups is precisely the class of capable finite abelian groups. This gives a complete description of the finite abelian groups B that satisfy the condition on the lattice of subgroups given above. The case of nonabelian groups seems much harder and not much is known at the 2000 Mathematics Subject Classification. 20K01, 20D30. The work presented here is partially supported by NSF/DMS-0805932.