Group with two conjugacy classes

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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.

Relation with other properties

Stronger properties

• Torsion-free group with two conjugacy classes

Weaker properties

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