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
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
|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|