Difference between revisions of "Characteristicity is transitive"

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A [[characteristic subgroup]] of a [[characteristic subgroup]] is characteristic in the whole group.
 
A [[characteristic subgroup]] of a [[characteristic subgroup]] is characteristic in the whole group.
  
===Symbolic statement===
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===Statement with symbols===
  
 
Let <math>H</math> be a [[characteristic subgroup]] of <math>K</math>, and <math>K</math> a characteristic subgroup of <math>G</math>. Then, <math>H</math> is a characteristic subgroup of <math>G</math>.
 
Let <math>H</math> be a [[characteristic subgroup]] of <math>K</math>, and <math>K</math> a characteristic subgroup of <math>G</math>. Then, <math>H</math> is a characteristic subgroup of <math>G</math>.
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* [[Characteristicity is transitive in Lie rings]]
 
* [[Characteristicity is transitive in Lie rings]]
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* [[Characteristicity is transitive in any variety of algebras]]
 
* [[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.  
 
* [[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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* [[Automorph-conjugacy is transitive]]
 
* [[Automorph-conjugacy is transitive]]
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==Definitions used==
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===Characteristic subgroup===
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{{further|[[Characteristic subgroup]]}}
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A subgroup <math>H</math> of a group <math>G</math> is termed a characteristic subgroup if whenever <math>\sigma</math> is an automorphism of <math>G</math>, <math>\sigma</math> restricts to an automorphism of <math>H</math>.
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This is written using the [[function restriction expression]]:
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Automorphism <math>\to</math> Automorphism
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In other words, every automorphism of the whole group restricts to an automorphism of the subgroup.
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===Transitive subgroup property==
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{{further|[[Transitive subgroup property]]}}
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A subgroup property <math>p</math> is termed transitive if whenever <math>H \le K \le G</math> are groups such that <math>H</math> satisfies property <math>p</math> in <math>K</math> and <math>K</math> satisfies property <math>p</math> in <math>G</math>, <math>H</math> also satisfies property <math>p</math> in <math>G</math>.
  
 
==Proof==
 
==Proof==
{{fillin}}
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===Hands-on proof===
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'''Given''': A group <math>G</math> with a characteristic subgroup <math>K</math>. <math>H</math> is a characteristic subgroup of <math>G</math>.
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'''To prove''': <math>H</math> is a characteristic subgroup of <math>G</math>: for any automorphism <math>\sigma</math> of <math>G</math>, <math>\sigma</math> restricts to an automorphism of <math>H</math>.
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'''Proof''':
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# <math>\sigma(K) = K</math>, and <math>\sigma</math> restricts to an automorphism of <math>K</math>: This follows from the fact that <math>K</math> is characteristic in <math>G</math>.
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# Let <math>\sigma'</math> be the restriction of <math>\sigma</math> to <math>K</math>. Then, <math>\sigma'</math> is an automorphism of <math>K</math> by step (1).
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# <math>\sigma'(H) = H</math>, and <math>\sigma'</math> restricts to an automorphism of <math>H</math>: This follows from the fact that <math>H</math> is characteristic in <math>K</math>.
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# Thus, <math>\sigma(H) = H</math>.
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{{frexp metaproperty satisfaction}}
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{{proof generalizes|[[balanced implies transitive]]}}
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The idea behind this proof is to observe that characteristicity can be written as the balanced subgroup property:
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Automorphism <math>\to</math> Automorphism
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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==
 
==References==

Revision as of 20:22, 20 October 2008

DIRECT: The fact or result stated in this article has a trivial/direct/straightforward proof provided we use the correct definitions of the terms involved
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VIEW: Survey articles about this
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


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 H be a characteristic subgroup of K, and K a characteristic subgroup of G. Then, H is a characteristic subgroup of G.

Related facts

Close relation with normality

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

Other related facts

Definitions used

Characteristic subgroup

Further information: Characteristic subgroup

A subgroup H of a group G is termed a characteristic subgroup if whenever \sigma is an automorphism of G, \sigma restricts to an automorphism of H.

This is written using the function restriction expression:

Automorphism \to 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 p is termed transitive if whenever H \le K \le G are groups such that H satisfies property p in K and K satisfies property p in G, H also satisfies property p in G.

Proof

Hands-on proof

Given: A group G with a characteristic subgroup K. H is a characteristic subgroup of G.

To prove: H is a characteristic subgroup of G: for any automorphism \sigma of G, \sigma restricts to an automorphism of H.

Proof:

  1. \sigma(K) = K, and \sigma restricts to an automorphism of K: This follows from the fact that K is characteristic in G.
  2. Let \sigma' be the restriction of \sigma to K. Then, \sigma' is an automorphism of K by step (1).
  3. \sigma'(H) = H, and \sigma' restricts to an automorphism of H: This follows from the fact that H is characteristic in K.
  4. Thus, \sigma(H) = H.

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 \to 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, 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)