Group in which any two elements generating the same cyclic subgroup are automorphic

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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 in which any two elements generating the same cyclic subgroup are automorphic is a group G satisfying the following equivalent conditions:

  1. g,h \in G are such that \langle g \rangle = \langle h \rangle, then there is an automorphism of G sending g to h.
  2. There exists a group K containing G as a normal subgroup such that all elements of G are rational elements of K.

Relation with other properties

Stronger properties

Weaker properties

Facts

Metaproperties

Characteristic subgroups

This group property is characteristic subgroup-closed: any characteristic subgroup of a group with the property, also has the property
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Quotients

This group property is quotient-closed, viz., any quotient of a group satisfying the property also has the property
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