Characteristicity is transitive

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This article gives the statement, and possibly proof, of a subgroup property (i.e., characteristic subgroup) satisfying a subgroup metaproperty (i.e., transitive subgroup property)
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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.

Symbolic statement

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

Close relation with normality

• Normality is not transitive: A normal subgroup of a normal subgroup need not be normal.
• Characteristic of normal implies normal
• Left transiter of normal is characteristic: If $H\le G$ is a subgroup such that whenever $G$ is normal in $K$, so is $H$, then $H$ is characteristic in $K$.

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:

• Full characteristicity is transitive
• Central factor is transitive

Analogues in other structures

• Characteristicity is transitive in Lie rings
• Derivation-invariance is transitive: For some purposes, the property of being a derivation-invariant Lie subring is the Lie analogue of the property of being a characteristic subgroup.

Other related facts

• Automorph-conjugacy is transitive

Proof

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References

Textbook references

• Abstract Algebra by David S. Dummit and Richard M. Foote, 10-digit ISBN 0471433349, 13-digit ISBN 978-0471433347, ^{More info}, Page 135, Page 137 (Problem 8(b))
• Groups and representations by Jonathan Lazare Alperin and Rowen B. Bell, ISBN 0387945261, ^{More info}, Page 17, Lemma 4
• A Course in the Theory of Groups by Derek J. S. Robinson, ISBN 0387944613, ^{More info}, Page 28, Characteristic and Fully invariant subgroups, 1.5.6(ii)
• Nilpotent groups and their automorphisms by Evgenii I. Khukhro, ISBN 3110136724, ^{More info}, Page 4, Section 1.1 (passing mention)