Group in which every automorphism is class-preserving

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This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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Definition

A group is termed a group in which every automorphism is class-preserving if it satisfies the following equivalent conditions:

  1. Every automorphism of the group is a class-preserving automorphism, i.e., it sends each element to within its conjugacy class.
  2. Every element of the group is automorph-conjugate, i.e., if two elements of the group are related by an automorphism, then they are in fact conjugate elements, i.e., they are in the same conjugacy class.

Relation with other properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
complete group Group in which every automorphism is inner|FULL LIST, MORE INFO
group in which every automorphism is inner |FULL LIST, MORE INFO