Group in which every automorphism is inner

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This article defines a group property: a property that can be evaluated to true/false for any given group
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RANDOM GROUP PROPERTY: Complete group: A group for which the natural homomorphism to its automorphism group is an isomorphism, i.e., a centerless group for which every automorphism is inner.

Contents

Definition

A group in which every automorphism is inner is a group with the property that every automorphism of the group is an inner automorphism, i.e., can be expressed via conjugation by an element.

Formalisms

In terms of the supergroup property collapse operator

This group property can be defined in terms of the collapse of two subgroup properties in the following sense. Whenever the given group is embedded as a subgroup satisfying the first subgroup property (normal subgroup), in some bigger group, it also satisfies the second subgroup property (central factor), and vice versa.
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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 (inner automorphism), and vice versa.
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Relation with other properties

Stronger properties

Weaker properties

Facts about Group in which every automorphism is innerRDF feed
Defining ingredientNormal subgroup  +, Central factor  +, Automorphism  +, and Inner automorphism  +
Stronger thanGroup in which every normal subgroup is characteristic  +, Group in which every subgroup is automorph-conjugate  +, Group in which every automorphism is class-preserving  +, and Group in which every automorphism is normal-extensible  +
Weaker thanComplete group  +
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