Group in which every endomorphism is trivial or injective

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.