# 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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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 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