# Preserves conjugacy classes for a generating set implies IA

From Groupprops

This article gives the statement and possibly, proof, of an implication relation between two automorphism properties. That is, it states that every automorphism satisfying the first automorphism property (i.e., automorphism that preserves conjugacy classes for a generating set) must also satisfy the second automorphism property (i.e., IA-automorphism)

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## Statement

Suppose is a group, is a generating set for , and is an automorphism of such that is in the conjugacy class of for all . Then, is an IA-automorphism of , i.e., the automorphism induced by on the abelianization of (the quotient group of by its derived subgroup ) is the identity map.

## Proof

**Given**: A group , a generating set for , an automorphism of such that, for every , there exists for which . is the derived subgroup of and is the quotient map to the abelianization .

**To prove**: For every ,

**Proof**: The proof has two steps:

- We first show the statement to be proved for all elements of the generating set .
- We then extend to all of using the fact that both and are homomorphisms.