# Group in which every endomorphism is trivial or an automorphism

From Groupprops

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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

View a complete list of group propertiesVIEW RELATED:

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

- Finite simple group:
`For full proof, refer: Finite simple implies every endomorphism is trivial or an automorphism` - Finite quasisimple group:
`For full proof, refer: Finite quasisimple implies every endomorphism is trivial or an automorphism` - Simple co-Hopfian group:
`For full proof, refer: Simple co-Hopfian implies every endomorphism is trivial or an automorphism`