IA-automorphism

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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 |


This article is about a definition in group theory that is standard among the group theory community (or sub-community that dabbles in such things) but is not very basic or common for people outside.
VIEW: Definitions built on this | Facts about this: (facts closely related to IA-automorphism, all facts related to IA-automorphism) |Survey articles about this | Survey articles about definitions built on this
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View a list of other standard non-basic definitions

History

The term IA-automorphism was coined by Seymour Bachmuth in his paper Automorphisms of free metabelian groups.

Definition

Symbol-free definition

An automorphism of a group is termed an IA-automorphism if it satisfies the following equivalent conditions:

  1. It induces the identity map on the abelianization of the group
  2. It takes each element to within its coset for the derived subgroup
  3. It induces the identity map on each of the quotient groups between successive members of the lower central series.

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
inner automorphism conjugation by a group element Automorphism that preserves conjugacy classes for a generating set, Class-preserving automorphism, Locally inner automorphism|FULL LIST, MORE INFO
locally inner automorphism effect on any finite subset mimics an inner automorphism Automorphism that preserves conjugacy classes for a generating set, Class-preserving automorphism|FULL LIST, MORE INFO
class-preserving automorphism sends every element to within its conjugacy class Class-preserving implies IA IA not implies class-preserving Automorphism that preserves conjugacy classes for a generating set|FULL LIST, MORE INFO
automorphism that preserves conjugacy classes for a generating set there exists a generating set all of whose elements are sent to conjugates by the automorphism Preserves conjugacy classes for a generating set implies IA (not obvious, may not even be true??) |FULL LIST, MORE INFO

Weaker properties

Metaproperties

Metaproperty Satisfied? Proof Statement with symbols
group-closed automorphism property Yes For any group G, the group of IA-automorphisms of G forms a subgroup of the automorphism group of G.

Facts

Related group properties

References

Journal references

  • Automorphisms of free metabelian groups by Seymour Bachmuth, Transactions of the AMS, 1965JSTOR linkMore info

Textbook references

  • Combinatorial Methods: Free Groups, Polynomials, and Free Algebras (CMS Books in Mathematics) by Vladimir Shpilrain, Alexander A. Mikhalev, and Jie-Tai Yu, ISBN 0387405623, More info, Page 21, Section 2.1 (Nielsen's commutator test) (definition given parenthetically)

External links

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