IA-automorphism

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

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.

Weaker properties

 * Automorphism

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

Textbook references

 * , Page 21, Section 2.1 (Nielsen's commutator test) (definition given parenthetically)