The Group Properties Wiki (pre-alpha)

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This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof.
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RANDOM SUBGROUP PROPERTY: Automorph-conjugate subgroup: A subgroup of a group that is conjugate to any automorphic subgroup. Any characteristic subgroup is automorph-conjugate.

Contents 1 Definition 2 Relation with other properties 2.1 Stronger properties 2.2 Weaker properties 2.3 Conjunction with other properties

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Definition

A subgroup of a group is termed a join of automorph-conjugate subgroups or AC-generated if it is generated by a collection of automorph-conjugate subgroups, viz it is a join of automorph-conjugate subgroups.

Relation with other properties

Stronger properties

• Automorph-conjugate subgroup
• Hall subgroup

Weaker properties

• Closure-characteristic subgroup

Conjunction with other properties

Any normal subgroup that is a join of automorph-conjugate subgroups is characteristic. Thus, this subgroup property is normal-to-characteristic