Preserves conjugacy classes for a generating set implies IA

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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 G is a group, S is a generating set for G, and \sigma is an automorphism of G such that \sigma(x) is in the conjugacy class of x for all x \in S. Then, \sigma is an IA-automorphism of G, i.e., the automorphism induced by \sigma on the abelianization of G (the quotient group G/G' of G by its derived subgroup G' = [G,G]) is the identity map.

Proof

Given: A group G, a generating set S for G, an automorphism \sigma of G such that, for every x \in S, there exists g \in G for which \sigma(x) = gxg^{-1}. G' is the derived subgroup of G and \varphi: G \to G/G' is the quotient map to the abelianization G/G'.

To prove: For every u \in G, \varphi(u) = \varphi(\sigma(u))

Proof: The proof has two steps:

  1. We first show the statement to be proved for all elements of the generating set S.
  2. We then extend to all of G using the fact that both \varphi and \varphi \circ \sigma are homomorphisms.