IA-automorphism
From Groupprops
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:
- It induces the identity map on the abelianization of the group
- It takes each element to within its coset for the derived subgroup
- 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
Weaker properties
Metaproperties
Metaproperty | Satisfied? | Proof | Statement with symbols |
---|---|---|---|
group-closed automorphism property | Yes | For any group , the group of IA-automorphisms of forms a subgroup of the automorphism group of . |
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
References
Journal references
- Automorphisms of free metabelian groups by Seymour Bachmuth, Transactions of the AMS, 1965^{JSTOR link}^{More 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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