The Group Properties Wiki (pre-alpha)

TIP: Having trouble locating the wiki page about a given fact? Get tips

ABOUT US: We use a Creative Commons license. All our content is free to reuse, with attribution. Learn more

ALSO CHECK OUT: Commalg: The Commutative Algebra Wiki

From Groupprops

Contents 1 Statement 1.1 Property-theoretic statement 1.2 Verbal statement 1.3 Statement with symbols 2 Related facts 2.1 Close relation with normality 2.2 Generalizations 2.3 Analogues in other structures 2.4 Other related facts 3 Definitions used 3.1 Characteristic subgroup 4 =Transitive subgroup property 5 Proof 5.1 Hands-on proof 6 References 6.1 Textbook references

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
View other results with direct proofs

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 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

• 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 G is normal in K, so is H, then H is characteristic in K.

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:

• Full characteristicity is transitive
• Central factor is transitive

Analogues in other structures

• Characteristicity is transitive in Lie rings
• 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.

Other related facts

• Automorph-conjugacy is transitive

Definitions used

Characteristic subgroup

Further information: Characteristic subgroup

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

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 p is termed transitive if whenever 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 σ of G, σ restricts to an automorphism of H.

Proof:

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

Template:Frexp metaproperty satisfaction

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, ISBN 0471433349, 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}