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 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)