On capability of finite abelian groups

TL;DR

This paper offers a new subgroup lattice-based characterization of capable finite abelian groups, expanding understanding beyond Baer's classical invariant factor condition.

Contribution

It introduces a novel criterion involving subgroup families and quotient properties, providing an alternative to Baer's invariant factor approach.

Findings

Characterization of capable finite abelian groups via subgroup families.

Equivalent conditions involving subgroup coverage and quotient isomorphism.

Broader criteria for capability in finite abelian groups.

Abstract

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.