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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Let be a characteristic subgroup of , and a characteristic subgroup of . Then, is a characteristic subgroup of .
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 .
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.
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- 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)