Group whose outer automorphism group is cyclic
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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A group whose outer automorphism group is cyclic is defined in the following equivalent ways:
- The outer automorphism group of the group is a cyclic group.
- The inner automorphism group is a 1-completed subgroup of the automorphism group, i.e., the automorphism group can be generated by the inner automorphism group and one element.
Relation with other properties
Conjunction with other properties
We are often interested in this property in the context of finite groups: finite group whose outer automorphism group is cyclic.
|Property||Meaning||Proof of implication||Proof of strictness (reverse implication failure)||Intermediate notions|
|Group in which every automorphism is inner||outer automorphism group is trivial|
|Complete group||centerless, every automorphism is inner|
|Group whose inner automorphism group is maximal in automorphism group|