Group admitting a partition into abelian subgroups

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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 properties
VIEW 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