Injective homomorphism

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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