This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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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.
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).
Relation with other properties
This group property is direct product-closed, viz., the direct product of an arbitrary (possibly infinite) family of groups each having the property, also has the property
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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.