Characteristic subgroup of finite group

This article describes a property that arises as the conjunction of a subgroup property: characteristic subgroup with a group property imposed on the ambient group: finite group
View a complete list of such conjunctions | View a complete list of conjunctions where the group property is imposed on the subgroup

Definition

A subgroup of a group is termed a characteristic subgroup of finite group if it satisfies the following equivalent conditions:

1. The whole group is a finite group and the subgroup is a characteristic subgroup of it.
2. The whole group is a finite group and the subgroup is a strictly characteristic subgroup (i.e., invariant under all surjective endomorphisms) of it.
3. The whole group is a finite group and the subgroup is an injective endomorphism-invariant subgroup of it.
4. The whole group is a finite group and the subgroup is a purely definable subgroup of it, i.e,, the subgroup is definable in the first-order theory of the group.
5. The whole group is a finite group and the subgroup is an elementarily characteristic subgroup of it, i.e., there is no other elementarily equivalently embedded subgroup.
6. The whole group is a finite group and the subgroup is a monadic second-order characteristic subgroup of it.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
isomorph-free subgroup of finite group
fully invariant subgroup of finite group
normal Sylow subgroup
normal Hall subgroup
characteristic subgroup of group of prime power order

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
normal subgroup of finite group
finite characteristic subgroup
finite normal subgroup