Surjective homomorphism of groups

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Definition

Suppose G and H are groups. A set map \varphi:G \to H is termed a surjective homomorphism of groups from G to H if it satisfies the following:

  1. \varphi is a homomorphism of groups from G to H and \varphi is surjective as a set map.
  2. \varphi is a homomorphism of groups from G to H and it is an epimorphism in the category of groups.
  3. \varphi is a homomorphism of groups from G to H and it descends to an isomorphism of groups from the quotient group G/K to H where K is the kernel of \varphi.

Equivalence of definitions

Related notions