# Group that is the characteristic closure of a singleton subset

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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 is termed the **characteristic closure of a singleton subset** if it satisfies the following equivalent conditions:

- is not the union of all its proper characteristic subgroups.
- There exists an element such that the characteristic closure of (or equivalently, the characteristic closure of the cyclic subgroup ) in is the whole group .

## Relation with other properties

### Stronger properties

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

Cyclic group | generated by one element | Group that is the normal closure of a singleton subset|FULL LIST, MORE INFO | ||

Group that is the normal closure of a singleton subset | normal closure of a singleton subset | |FULL LIST, MORE INFO | ||

Characteristically simple group | no proper nontrivial characteristic subgroups | |FULL LIST, MORE INFO | ||

Simple group | no proper nontrivial normal subgroups | Characteristically simple group, Group that is the normal closure of a singleton subset|FULL LIST, MORE INFO | ||

Finite abelian group | finite and abelian | |FULL LIST, MORE INFO | ||

Free group | |FULL LIST, MORE INFO | |||

Reduced free group | |FULL LIST, MORE INFO |