# Group in which every characteristic submonoid is a subgroup

From Groupprops

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 a **group in which every characteristic submonoid is a subgroup** if it satisfies the following equivalent conditions:

- Every characteristic subset that is a submonoid under the group multiplication is a subgroup, and hence a characteristic subgroup, of the whole group.
- Every characteristic subset that is nonempty and is a subsemigroup under the group multiplication is a subgroup, and hence a characteristic subgroup, of the whole group.

## Facts

- Characteristic submonoid of group not implies subgroup shows that this property is not tautological.

## Relation with other properties

### Stronger properties

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

finite group | nonempty finite subsemigroup of group is subgroup | any infinite abelian group | |FULL LIST, MORE INFO | |

periodic group | every submonoid is a subgroup | |FULL LIST, MORE INFO | ||

abelian group | nonempty characteristic subsemigroup of abelian group is subgroup | any finite non-abelian group | Template:Group in which every characteristic submonoid is a subgroup | |

group in which every element is automorphic to its inverse | any finite group that fails this property, such as nontrivial semidirect product of Z7 and Z3 | |FULL LIST, MORE INFO |