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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]]