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 be a characteristic subgroup of , and a characteristic subgroup of . Then, is a characteristic subgroup of .
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 is a subgroup such that whenever is normal in , so is , then is characteristic in .
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:
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.
Proof
PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]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)