# Injective homomorphism

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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:

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

### Definition with symbols

Suppose is a homomorphism of groups. We say that is a **monomorphism** or an **injective homomorphism** if it satisfies the following equivalent conditions:

- is injective as a map of sets
- The kernel of the map, i.e. the subgroup of given by where is the identity element of , is the trivial subgroup of . In other words,
- 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`