On capability of finite abelian groups

Details On capability of finite abelian groups On capability of finite abelian groups

2009-02-24

arxiv.org/abs/0902.4089v1

Mathematics

Group Theory

Scientific paper

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 {H_i} of subgroups of G with trivial intersection, such that the union generates G and all the quotients G/H_i 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.

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