Characteristicity is transitive

Statement
Let $$H$$ be a characteristic subgroup of $$K$$, and $$K$$ a characteristic subgroup of $$G$$. Then, $$H$$ is a characteristic subgroup of $$G$$.

Close relation with normality
A normal subgroup is a subgroup that is invariant under all inner automorphisms.

Below, we take $$H \le K \le G$$, with $$H$$ the bottom group, $$K$$ the middle group, and $$G$$ the top group.

Generalizations
Balanced implies transitive: Any subgroup property that can be expressed as a balanced subgroup property is transitive. Characteristicity is a special case. Other special cases include:

Other related facts

 * Automorph-conjugacy is transitive
 * SQ-dual::Characteristicity is quotient-transitive

Characteristic subgroup
A subgroup $$H$$ of a group $$G$$ is termed a characteristic subgroup if whenever $$\sigma$$ is an automorphism of $$G$$, $$\sigma$$ restricts to an automorphism of $$H$$.

This is written using the function restriction expression:

Automorphism $$\to$$ Automorphism

In other words, every automorphism of the whole group restricts to an automorphism of the subgroup.

Transitive subgroup property
A subgroup property $$p$$ is termed transitive if whenever $$H \le K \le G$$ are groups such that $$H$$ satisfies property $$p$$ in $$K$$ and $$K$$ satisfies property $$p$$ in $$G$$, $$H$$ also satisfies property $$p$$ in $$G$$.

Hands-on proof
Given: A group $$G$$ with a characteristic subgroup $$K$$. $$H$$ is a characteristic subgroup of $$K$$. $$\sigma$$ is an automorphism of $$G$$.

To prove: $$\sigma(H) = H$$ and $$\sigma$$ restricts to an automorphism of $$H$$.

Proof:

The idea behind this proof is to observe that characteristicity can be written as the balanced subgroup property:

Automorphism $$\to$$ Automorphism

In other words, every automorphism of the big group restricts to an automorphism of the subgroup. Any balanced subgroup property is transitive, and this gives the proof.