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: Group satisfying normalizer condition: A group with no proper self-normalizing subgroup.

Contents 1 Definition 2 Formalisms 2.1 In terms of the subgroup property collapse operator 2.2 In terms of the automorphism property collapse operator

Definition

A group in which every subgroup is automorph-conjugate is a group satisfying the following equivalent conditions:

• Every subgroup of the group is an automorph-conjugate subgroup.
• Any two automorphic subgroups of the group are conjugate subgroups.
• Every automorphism of the group is a subgroup-conjugating automorphism.

Formalisms

In terms of the subgroup property collapse operator

This group property can be defined in terms of the collapse of two subgroup properties. In other words, a group satisfies this group property if and only if every subgroup of it satisfying the first property (subgroup) satisfies the second property (automorph-conjugate subgroup), and vice versa.
View other group properties obtained in this way

In terms of the automorphism property collapse operator

This group property can be defined in terms of the collapse of two automorphism properties. In other words, a group satisfies this group property if and only if every automorphism of it satisfying the first property (automorphism) satisfies the second property (subgroup-conjugating automorphism), and vice versa.
View other group properties obtained in this way