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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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Contents

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

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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