Fully invariant implies characteristic
This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., fully invariant subgroup) must also satisfy the second subgroup property (i.e., characteristic subgroup)
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Fully invariant subgroup
Further information: fully invariant subgroup
A subgroup of a group is termed fully invariant in if, for every endomorphism of , is contained in .
Further information: characteristic subgroup
A subgroup of a group is termed characteristic in if, for every automorphism of , is contained in .
- Characteristic not implies fully invariant
- Characteristic not implies fully invariant in finite abelian group
- Characteristic equals fully invariant in odd-order abelian group
For intermediate notions between characteristic subgroup and fully invariant subgroup, click here.
Invariance under restricted classes of endomorphisms
- Strictly characteristic subgroup is a subgroup that is invariant under all surjective endomorphisms. Every fully invariant subgroup is strictly characteristic, and every strictly characteristic subgroup is characteristic.
- Injective endomorphism-invariant subgroup is a subgroup that is invariant under all injective endomorphisms. Every fully invariant subgroup is strictly characteristic, and every strictly characteristic subgroup is characteristic.
The idea behind the proof is that since every automorphism is an endomorphism, invariance under all endomorphisms implies invariance under automorphisms.
Using function restriction expressions
This subgroup property implication can be proved by using function restriction expressions for the subgroup properties
View other implications proved this way |read a survey article on the topic
Since the left side of the function restriction expression for characteristicity is stronger while the right sides of both function restriction expressions are equal, the property of being characteristic is weaker than the property of being fully invariant.