# Group that is the characteristic closure of a singleton subset

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## Contents

BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]
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 $G$ is termed the characteristic closure of a singleton subset if it satisfies the following equivalent conditions:

1. $G$ is not the union of all its proper characteristic subgroups.
2. There exists an element $g \in G$ such that the characteristic closure of $\{ g \}$ (or equivalently, the characteristic closure of the cyclic subgroup $\langle g \rangle$) in $G$ is the whole group $G$.

## 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