# Group admitting a partition into abelian subgroups

From Groupprops

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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

## Definition

A **group admitting a partition into abelian subgroups** is defined as a group that can be expressed as a union of abelian subgroups (i.e., subgroups that are all abelian groups) of which any two have trivial intersection. In other words, it admits a Partition of a group (?) where all the parts are abelian subgroups.

Note that any nontrivial group with this property is a group admitting a nontrivial partition.

## Relation with other properties

### Stronger properties

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

Group admitting a partition into cyclic subgroups | can choose partition where all parts are cyclic | |FULL LIST, MORE INFO | ||

Cyclic group | generated by one element | Group admitting a partition into cyclic subgroups|FULL LIST, MORE INFO | ||

Abelian group | any two elements commute | |FULL LIST, MORE INFO | ||

Dihedral group | semidirect product of cyclic group, element acting via inverse map | Group admitting a partition into cyclic subgroups|FULL LIST, MORE INFO | ||

Generalized dihedral group | semidirect product of abelian group, element acting via inverse map | |FULL LIST, MORE INFO |