First group isomorphism theorem

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Note:
First isomorphism theorem

Where θ is an isomorphism.
Properties
something

Statement

Let (G,) and (H,) be groups. Let φ:GH be a group homomorphism, then[1]:

  • G/Ker(φ)Im(φ)
    • Explicitly we may state this as: there exists a group isomorphism between G/Ker(φ) and Im(φ).

Note: the special case of φ being surjective, then Im(φ)=H, so we see G/Ker(φ)H

Proof

Notes

References

  1. Jump up Abstract Algebra - Pierre Antoine Grillet