# Group in which every characteristic submonoid is a subgroup

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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## Definition

A group $G$ is termed a group in which every characteristic submonoid is a subgroup if it satisfies the following equivalent conditions:

1. Every characteristic subset that is a submonoid under the group multiplication is a subgroup, and hence a characteristic subgroup, of the whole group.
2. 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.

## 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