Injective homomorphism

This article is about a basic definition in group theory. The article text may, however, contain advanced material.
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Definition

QUICK PHRASES: injective homomorphism, homomorphism with trivial kernel, monic, monomorphism

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 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. For full proof, refer: Monomorphism iff injective in the category of groups