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 $K$.

Relation with other properties

Stronger properties

• Group with two conjugacy classes
• Group whose automorphism group is transitive on non-identity elements
• Group in which every element is order-conjugate
• Group in which every element is order-automorphic
• Rational group
• Finite abelian group

Weaker properties

• Group in which every element is automorphic to its inverse

Facts

• Normal subgroup of rational group implies any two elements generating the same cyclic subgroup are automorphic
• Alternating group implies any two elements generating the same cyclic subgroup are automorphic

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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