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

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 $K$.

Metaproperties

Characteristic subgroups

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

Quotients

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