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
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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