# Automorphism that preserves conjugacy classes for a generating set

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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## Definition

Suppose $G$ is a group and $\sigma$ is an automorphism of $G$.We say that $\sigma$ is an automorphism that preserves conjugacy classes for a generating set if the folllowing equivalent conditions hold:

1. There is a generating set $S$ for $G$ such that, for all $x \in S$, $\sigma(x)$ is in the conjugacy class of $x$, i.e., there exists $g \in G$ such that $\sigma(x) = gxg^{-1}$. Note that $g$ depends on $x$.
2. There is a symmetric subset $T$ of $G$ that is a generating set of $G$ and such that, for all $x \in T$, $\sigma(x)$ is in the conjugacy class of $x$, i.e., there exists $g \in G$ such that $\sigma(x) = gxg^{-1}$. Note that $g$ depends on $x$.
3. There is a normal subset $N$ of $G$ that is a generating set of $G$ and such that, for all $x \in N$, $\sigma(x)$ is in the conjugacy class of $x$, i.e., there exists $g \in G$ such that $\sigma(x) = gxg^{-1}$. Note that $g$ depends on $x$.

## Relation with other properties

### Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
inner automorphism an automorphism for which there exists a single element $g \in G$ such that the automorphism is the map $x \mapsto gxg^{-1}$ (obvious) (via class-preserving automorphism) |FULL LIST, MORE INFO
locally inner automorphism looks like an inner automorphism in its effect on any finite subset (obvious) (via class-preserving automorphism) |FULL LIST, MORE INFO
class-preserving automorphism sends every element to within its conjugacy class (obvious) preserves conjugacy classes for a generating set not implies class-preserving |FULL LIST, MORE INFO

### Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
IA-automorphism sends every element to within its coset of the derived subgroup, or equivalently, induces the identity automorphism on the abelianization preserves conjugacy classes for a generating set implies IA IA-automorphism|automorphism that preserves conjugacy classes for a generating set]]