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)
View all subgroup metaproperty satisfactions | View all subgroup metaproperty dissatisfactions |Get help on looking up metaproperty (dis)satisfactions for subgroup properties
Get more facts about characteristic subgroup |Get facts that use property satisfaction of characteristic subgroup | Get facts that use property satisfaction of characteristic subgroup|Get more facts about transitive subgroup property
Close relation with normality
Below, we take , with the bottom group, the middle group, and the top group.
|Statement||Change in assumption||Change in conclusion|
|Normality is not transitive||normal in , normal in||not normal in|
|Characteristic of normal implies normal||normal in||normal in|
|Left transiter of normal is characteristic||in such that if is normal in , is normal in||is characteristic in|
Analogues in other algebraic structures
Generalizations in the one-of-its-kind sense of the statement
|Second-order characteristic subgroup||no subgroup equivalent in the second-order theory of groups||Second-order characteristicity is transitive|
|Monadic second-order characteristic subgroup||no subgroup equivalent is the monadic second-order theory of groups||Monadic second-order characteristicity is transitive|
|Purely definable subgroup||definable in the pure theory of groups||Pure definability is transitive|
Further information: Characteristic subgroup
A subgroup of a group is termed a characteristic subgroup if whenever is an automorphism of , restricts to an automorphism of .
This is written using the function restriction expression:
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 is termed transitive if whenever are groups such that satisfies property in and satisfies property in , also satisfies property in .
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Given: A group with a characteristic subgroup . is a characteristic subgroup of . is an automorphism of .
To prove: and restricts to an automorphism of .
|Step no.||Assertion/construction||Facts used||Given data used||Previous steps used||Explanation|
|1||, and restricts to an automorphism of , that we call .||definition of characteristic subgroup||is characteristic in , is an automorphism of .||direct|
|2||, and restricts to an automorphism of||definition of characteristic subgroup||is characteristic in||Step (1)||direct|
|3||and restricts to an automorphism of .||Steps (1), (2)||[SHOW MORE]|
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:
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.
- 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