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

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:

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)