IA-automorphism

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)
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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.
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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			|FULL LIST, MORE INFO
locally inner automorphism	effect on any finite subset mimics an inner 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	|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

• Automorphism

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

• IA-automorphism group of finite p-group is p-group
• IA-automorphism group of finite nilpotent group has precisely the same prime factors of order as the derived subgroup

Related group properties

• Group in which every IA-automorphism is inner

References

Journal references

• Automorphisms of free metabelian groups by Seymour Bachmuth, Transactions of the AMS, 1965^{JSTOR 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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