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.
Statement with symbols
Let be a characteristic subgroup of , and a characteristic subgroup of . Then, is a characteristic subgroup of .
Related facts
Close relation with normality
A normal subgroup is a subgroup that is invariant under all inner automorphisms.
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 |
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 algebraic structures
Generalizations in the one-of-its-kind sense of the statement
Property | Meaning | Proof |
---|---|---|
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 |
Definitions used
Characteristic subgroup
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:
Automorphism Automorphism
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 .
Proof
Hands-on proof
Given: A group with a characteristic subgroup . is a characteristic subgroup of .
To prove: is a characteristic subgroup of : for any automorphism of , restricts to an automorphism of .
Proof:
- , and restricts to an automorphism of : This follows from the fact that is characteristic in .
- Let be the restriction of to . Then, is an automorphism of by step (1).
- , and restricts to an automorphism of : This follows from the fact that is characteristic in .
- Thus, .
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:
Automorphism Automorphism
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.
References
Textbook references
- 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}