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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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.
- Characteristic submonoid of group not implies subgroup shows that this property is not tautological.
Relation with other 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|