Group with two conjugacy classes
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 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
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