# Direct product of finite groups

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

## Definition

A group is termed a **direct product of finite groups** if it is isomorphic to the external direct product of a collection of (possibly infinitely many) finite groups.

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

Finite group | |FULL LIST, MORE INFO | |||

Direct power of a finite group | |FULL LIST, MORE INFO | |||

Direct product of finite groups with only finitely many isomorphism classes of direct factors | A direct product of finite groups where there are only finitely many isomorphism classes among the finite groups occurring as the factors in the direct product. | |FULL LIST, MORE INFO | ||

Periodic direct product of finite groups | A direct product of finite groups that is also a periodic group | |FULL LIST, MORE INFO |

### Weaker properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

Residually finite group | A subdirect product of finite groups | Profinite group|FULL LIST, MORE INFO |