Group in which every automorphism is inner

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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VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

Definition

A group in which every automorphism is inner is a group satisfying the following equivalent conditions:

• Every automorphism of the group is an inner automorphism, i.e., can be expressed via conjugation by an element.
• The outer automorphism group of the group is trivial.

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

• Complete group

Weaker properties

• Group in which every normal subgroup is characteristic
• Group in which every subgroup is automorph-conjugate
• Group in which every automorphism is class-preserving
• Group in which every automorphism is normal-extensible

Facts

• There do exist finite solvable groups for which every automorphism is inner. In fact, there exist finite solvable groups that are complete: they are centerless and every automorphism is inner. In fact, there can exist multiple non-isomorphic complete solvable groups of the same order. Further information: Complete and composition factor-equivalent not implies isomorphic
• For any finite nilpotent group other than the trivial group or the cyclic group of order two, there are outer automorphisms. Further information: Finite nilpotent and every automorphism is inner implies trivial or cyclic of order two