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

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 defining ingredient::rational elements of $$K$$.

Stronger properties

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

Weaker properties

 * Stronger than::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