Group with two conjugacy classes

Definition
A group with two conjugacy classes is a nontrivial group satisfying the following equivalent conditions:


 * All its non-identity element are conjugate.
 * The inner automorphism group acts transitively on the set of non-identity elements.
 * It has exactly two conjugacy classes of elements.

Stronger properties

 * Weaker than::Torsion-free group with two conjugacy classes

Weaker properties

 * Stronger than::Simple group
 * Stronger than::Group in which every element is order-conjugate
 * Stronger than::Rational group
 * Stronger than::Ambivalent group
 * Stronger than::Group having a class-inverting automorphism
 * Stronger than::Group whose automorphism group is transitive on non-identity elements
 * Stronger than::Group in which every element is order-automorphic
 * Stronger than::Group in which any two elements generating the same cyclic subgroup are automorphic
 * Stronger than::Group in which every element is automorphic to its inverse