# CIA-group

From Groupprops

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism

View a complete list of group propertiesVIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

## Contents

## History

### 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.

## Definition

### 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*).

## Relation with other properties

### Stronger properties

- Abelian group
- finite Hamiltonian group

### Weaker properties

### Opposite properties

## Metaproperties

### Direct products

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

View other direct product-closed group properties

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.