This article is about a basic definition in group theory. The article text may, however, contain advanced material.
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QUICK PHRASES: injective homomorphism, homomorphism with trivial kernel, monic, monomorphism
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