Group in which every characteristic submonoid is a subgroup

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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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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.

Facts

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