Finite group in which every endomorphism is trivial or an automorphism

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This article defines a property that can be evaluated for finite groups (and hence, a particular kind of group property)
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Definition

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

Relation with other properties

Stronger properties