# Characteristicity is transitive

## Statement

### Property-theoretic statement

The subgroup property of being characteristic satisfies the subgroup metaproperty of being transitive.

### Verbal statement

A characteristic subgroup of a characteristic subgroup is characteristic in the whole group.

### Statement with symbols

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

## Related facts

### 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:

## Definitions used

### Characteristic subgroup

Further information: 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

Further information: 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 $G$.

## Proof

### Hands-on proof

Given: A group $G$ with a characteristic subgroup $K$. $H$ is a characteristic subgroup of $G$.

To prove: $H$ is a characteristic subgroup of $G$: for any automorphism $\sigma$ of $G$, $\sigma$ restricts to an automorphism of $H$.

Proof:

1. $\sigma(K) = K$, and $\sigma$ restricts to an automorphism of $K$: This follows from the fact that $K$ is characteristic in $G$.
2. Let $\sigma'$ be the restriction of $\sigma$ to $K$. Then, $\sigma'$ is an automorphism of $K$ by step (1).
3. $\sigma'(H) = H$, and $\sigma'$ restricts to an automorphism of $H$: This follows from the fact that $H$ is characteristic in $K$.
4. Thus, $\sigma(H) = H$.

### Function restriction expression metaproperty satisfaction

This proof of a subgroup property satisfying a subgroup metaproperty relies on the nature of a function restriction expression for the subgroup property.

This proof method generalizes to the following results: balanced implies transitive

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.

## References

### Textbook references

• Abstract Algebra by David S. Dummit and Richard M. Foote, 10-digit ISBN 0471433349, 13-digit ISBN 978-0471433347, Page 137, Problem 8(b), More info
• Groups and representations by Jonathan Lazare Alperin and Rowen B. Bell, ISBN 0387945261, Page 17, Lemma 4, More info
• A Course in the Theory of Groups by Derek J. S. Robinson, ISBN 0387944613, Page 28, Section 1.5 (Characteristic and Fully invariant subgroups), 1.5.6(ii), More info
• Nilpotent groups and their automorphisms by Evgenii I. Khukhro, ISBN 3110136724, Page 4, Section 1.1, (passing mention)More info