Groupprops, The Group Properties Wiki (pre-alpha)

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This article defines a group property: a property that can be evaluated to true/false for any given group
View a complete list of group properties
VIEW RELATED:
RANDOM GROUP PROPERTY: Simple group: A nontrivial group having no proper nontrivial normal subgroup.

Contents 1 Definition 2 Relation with other properties 2.1 Stronger properties 2.2 Weaker properties

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