Injective homomorphism

Symbol-free definition
A homomorphism of groups is termed a monomorphism or an injective homomorphism if it satisfies the following equivalent conditions:


 * 1) It is injective as a map of sets
 * 2) Its Defining ingredient::kernel (the inverse image of the identity element) is trivial
 * 3) It is a monomorphism (in the category-theoretic sense) with respect to the category of groups

Definition with symbols
Suppose $$\varphi:G \to K$$ is a homomorphism of groups. We say that $$\varphi$$ is a monomorphism or an injective homomorphism if it satisfies the following equivalent conditions:


 * 1) $$\varphi$$ is injective as a map of sets
 * 2) The kernel of the map, i.e. the subgroup of $$G$$ given by $$\varphi^{-1}(\{ e \})$$ where $$e$$ is the identity element of $$K$$, is the trivial subgroup of $$G$$. In other words, $$\varphi^{-1}(\{ e \} = \{ e \}$$
 * 3) $$\varphi$$ is a monomorphism (in the category-theoretic sense) with respect to the category of groups.

Equivalence of definitions

 * Clearly (1) implies (2). The reverse implication follows from the first isomorphism theorem.
 * (1) equals (3) as well.