# Difference between revisions of "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
xharacteristic subgroup of group of prime power order

### Weaker properties

property quick description proof of implication proof of strictness (reverse implication failure) intermediate notions
normal subgroup of finite group
finite characteristic subgroup
finite normal subgroup