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From Groupprops

Contents 1 History 2 Definition 2.1 Symbol-free definition 2.2 Definition with symbols 3 Formalisms 3.1 Second-order description 3.2 Function restriction expression 4 Relation with other properties 4.1 Stronger properties 4.2 Weaker properties 5 Metaproperties 5.1 Transitivity 5.2 Trimness 5.3 Intersection-closedness 5.4 Join-closedness 6 References 6.1 Textbook references 7 External links 7.1 Definition links

This article is about a standard (though not very rudimentary) definition in group theory. The article text may, however, contain more than just the basic definition
View a complete list of semi-basic definitions on this wiki

This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof.
View a complete list of subgroup properties

This is a variation of characteristicity
View a complete list of variations of characteristicity OR read a survey article on varying characteristicity

History

This term was introduced by: Levi
View a complete list of terminology introduced by Levi

The concept was introduced by Levi in 1933 under the German name vollinvariant (translating to fully invariant). Both the terms fully characteristic and fully invariant are now in vogue.

Definition

Symbol-free definition

A subgroup of a group is termed fully characteristic or fully invariant if it is invariant under all endomorphisms of the whole group.

Definition with symbols

A subgroup H of a group G is termed fully characteristic if, for any endomorphism φ of G:

Formalisms

BEWARE! This section of the article uses terminology local to the wiki, possibly without giving a full explanation of the terminology used (though efforts have been made to clarify terminology as much as possible within the particular context)

Second-order description

This subgroup property is a second-order subgroup property, viz it has a second-order description in the theory of groups

Function restriction expression

This subgroup property can be expressed by means of the function restriction formalism, viz there is a function restriction expression for it.
View other properties expressible in this formalism OR View the function restriction formalism chart for a graphic placement of this property

The property of being fully characteristic can be expressed in terms of the function restriction formalism in the following ways:

• As the invariance property with respect to the function property of being an endomorphism:

Fully characteristic = Endomorphism Function

In other words, a subgroup is fully characteristic if and only if every endomorphism of the whole group restricts to a function on the subgroup (that is, takes the subgroup to within itself).

• As the balanced subgroup property (function restriction formalism) with respect to the function property of being an endomorphism:

Fully characteristic = Endomorphism Endomorphism

In other words, a subgroup is fully characteristic if and only if every endomorphism of the whole group restricts to an endomorphism of the subgroup.

Relation with other properties

Stronger properties

• Verbal subgroup: For full proof, refer: verbal implies fully characteristic
• Bound-word subgroup: For full proof, refer: bound-word implies fully characteristic

Weaker properties

• Strictly characteristic subgroup
• Characteristic subgroup
• Normal subgroup

Metaproperties

Transitivity

This subgroup property is transitive: a subgroup with this property in a subgroup with this property, also has this property
View a complete list of transitive subgroup properties

The property of being fully characteristic is a transitive subgroup property. That is, a fully characteristic subgroup of a fully characteristic subgroup is fully characteristic. This is because full characteristicity is a balanced subgroup property in the function restriction formalism. For full proof, refer: Full characteristicity is transitive

Trimness

This subgroup property is trim -- it is both trivially true (true for the trivial subgroup) and identity-true (true for a group as a subgroup of itself)
View all trim subgroup properties OR view trivially true subgroup properties OR view identity-true subgroup properties

The property of being fully characteristic is trim, in the sense:

• The trivial subgroup is always fully characteristic
• The whole group is always fully characteristic in itself

Intersection-closedness

This subgroup property is intersection-closed: an arbitrary (nonempty) intersection of subgroups with this property, also has this property
View a complete list of intersection-closed subgroup properties

The property of being fully characteristic is intersection-closed. That is, the intersection of a family of fully characteristic subgroups is fully characteristic. This follows from the fact that the property of being fully characteristic is an invariance property.For full proof, refer: Full characteristicity is intersection-closed

Join-closedness

This subgroup property is join-closed: an arbitrary (nonempty) join of subgroups with this property, also has this property
View a complete list of join-closed subgroup properties

The property of being fully characteristic is join-closed, viz an arbitrary join (subgroup generated) of fully characteristic subgroups is fully characteristic. This follows from the fact that full characteristicity is an endo-invariance property (an invariance property with respect to certain kinds of endomorphisms).

For full proof, refer: Full characteristicity is join-closed

References

Textbook references

• A Course in the Theory of Groups by Derek J. S. Robinson^{More info}, Page 28, Characteristic and fully invariant subgroups

External links

Search for "fully+characteristic+subgroup"OR"fully+invariant+subgroup" on the World Wide Web:
Scholarly articles: Google Scholar, JSTOR
Books: Google Books, Amazon
This wiki: Internal search, Google site search
Encyclopaedias: Wikipedia (or using Google), Citizendium
Math resource pages:Mathworld, Planetmath, Springer Online Reference Works
Math wikis: Topospaces, Diffgeom, Commalg, Noncommutative
Discussion fora: Mathlinks, Google Groups
The web: Google, Yahoo
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Definition links

• Wikipedia page
• Planetmath page