# Group in which every endomorphism is trivial or injective

From Groupprops

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism

View a complete list of group propertiesVIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

## Contents

## Definition

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

- 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).
- 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 |