Group in which every endomorphism is trivial or injective

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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Definition

A group in which every endomorphism is trivial or injective is a (usually nontrivial) group satisfying the following equivalent conditions:

1. Every endomorphism is either the trivial map (i.e., it maps everything to the identity map) or is an injective endomorphism (i.e., its kernel is trivial).
2. It has no proper nontrivial endomorphic kernel.

Note that whether the trivial group is included or not is a matter of convention. When thinking of it in terms of a simple group operator, we exclude the trivial group.

Formalisms

In terms of the simple group operator

This property is obtained by applying the simple group operator to the property: endomorphic kernel
View other properties obtained by applying the simple group operator

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Simple group	nontrivial, no proper nontrivial normal subgroup			|FULL LIST, MORE INFO
Splitting-simple group	nontrivial, no proper nontrivial retract (equivalently, no proper nontrivial complemented normal subgroup)			|FULL LIST, MORE INFO
Group in which every endomorphism is trivial or an automorphism	nontrivial, every endomorphism is either the trivial map or an automorphism			|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Hopfian group	every surjective endomorphism is an automorphism			|FULL LIST, MORE INFO