Class-preserving automorphism

From Groupprops
Jump to: navigation, search
This article defines a term that has been used or referenced in a journal article or standard publication, but may not be generally accepted by the mathematical community as a standard term.[SHOW MORE]
This article defines an automorphism property, viz a property of group automorphisms. Hence, it also defines a function property (property of functions from a group to itself)
View other automorphism properties OR View other function properties
This is a variation of inner automorphism|Find other variations of inner automorphism |

Definition

An automorphism of a group is termed a class-preserving automorphism or class automorphism if it takes each element to within its conjugacy class. In symbols, an automorphism \sigma of a group G is termed a class automorphism or class-preserving automorphism if for every g in G, there exists an element h such that \sigma(g) = hgh^{-1}. The choice of h may depend on g.

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Inner automorphism conjugation by an element inner implies class-preserving class-preserving not implies inner Hall-extensible automorphism, Locally inner automorphism|FULL LIST, MORE INFO
Locally inner automorphism preserves conjugacy classes of finite tuples locally inner implies class-preserving class-preserving not implies locally inner |FULL LIST, MORE INFO

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
automorphism that preserves conjugacy classes for a generating set there exists a generating set all of whose elements are sent to within their conjugacy class by the automorphism (obvious) preserves conjugacy classes for a generating set not implies class-preserving |FULL LIST, MORE INFO
IA-automorphism induces identity map on abelianization class-preserving implies IA IA not implies class-preserving Automorphism that preserves conjugacy classes for a generating set|FULL LIST, MORE INFO
normal automorphism sends each normal subgroup to itself class-preserving implies normal normal not implies class-preserving Extended class-preserving automorphism, Rational class-preserving automorphism|FULL LIST, MORE INFO
weakly normal automorphism sends each normal subgroup to a subgroup of itself (via normal) (via normal) Extended class-preserving automorphism, Normal automorphism, Rational class-preserving automorphism|FULL LIST, MORE INFO
extended class-preserving automorphism sends each element to conjugate or conjugate of inverse extended class-preserving not implies class-preserving |FULL LIST, MORE INFO
rational class-preserving automorphism sends each element to conjugate of element generating same cyclic group |FULL LIST, MORE INFO
center-fixing automorphism fixes every element of center class-preserving implies center-fixing center-fixing not implies class-preserving |FULL LIST, MORE INFO

Related properties

Facts

Metaproperties

Group-closedness

This automorphism property is group-closed: it is closed under the group operations on automorphisms (composition, inversion and the identity map). It follows that the subgroup comprising automorphisms with this property, is a normal subgroup of the automorphism group
View a complete list of group-closed automorphism properties

Clearly, a product of class automorphisms is a class automorphism, and the inverse of a class automorphism is a class automorphism. Thus, the class automorphisms form a group which sits as a subgroup of the automorphism group. Moreover, this subgroup contains the group of inner automorphisms, and is a normal subgroup inside the automorphism group.

Direct product-closedness

This automorphism property is direct product-closed
View a complete list of direct product-closed automorphism properties

Let G_1 and G_2 be groups and \sigma_1, \sigma_2 be class automorphisms of G_1, G_2 respectively. Then, \sigma_1 \times \sigma_2 is a class automorphism of G_1 \times G_2.

Here, \sigma_1 \times \sigma_2 is the automorphism of G_1 \times G_2 that acts as \sigma_1 on the first coordinate and \sigma_2 on the second.

References

External links

Search for "class+preserving+automorphism" 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, Windows Live
Learn more about using the Searchbox OR provide your feedback