Fundamental Theorem On Homomorphisms - Group Theoretic Version

Group Theoretic Version

Given two groups G and H and a group homomorphism f : GH, let K be a normal subgroup in G and φ the natural surjective homomorphism GG/K (where G/K is a quotient group). If K is a subset of ker(f) then there exists a unique homomorphism h:G/KH such that f = h φ.

The situation is described by the following commutative diagram:

By setting K = ker(f) we immediately get the first isomorphism theorem.

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