CIA-group

Origin
The notion of CIA-group was introduced in a paper titled Groups with a finite covering by isomorphic Abelian subgroups authored by Tuval Foguel and Matthew Ragland.

Symbol-free definition
A group is said to be a CIA-group if it can be expressed as a finite union of isomorphic Abelian groups (CIA stands for covered by isomorphism Abelian).

Stronger properties

 * Abelian group
 * finite Hamiltonian group

Weaker properties

 * Central-by-finite group
 * Finitely covered group

Opposite properties

 * Simple group
 * Frobenius group

Metaproperties
Any direct product of CIA-groups is a CIA-group. The covering Abelian groups are in fact simply the direct products of the covering Abelian subgroups for the two direct factors.