Group in which every endomorphism is trivial or an automorphism

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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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The version of this for finite groups is at: finite group in which every endomorphism is trivial or an automorphism

Definition

A group in which every endomorphism is trivial or an automorphism is a (typically, nontrivial) group for which every endomorphism is either the trivial map (sending all group elements to the identity element) or is an automorphism.

Whether the trivial group is included or not is a matter of convention. We sometimes exclude it when comparing with other simple group-type properties.

Relation with other properties

Stronger properties

Weaker properties