Fundamental theorem on homomorphisms

In abstract algebra, the fundamental theorem on homomorphisms, also known as the fundamental homomorphism theorem, relates the structure of two objects between which a homomorphism is given, and of the kernel and image of the homomorphism.

The homomorphism theorem is used to prove the isomorphism theorems.

Group theoretic version
Diagram of the fundamental theorem on homomorphisms where f is a homomorphism, N is a normal subgroup of G and e is the identity element of G

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

In other words, the natural projection φ is universal among homomorphisms on G that map K to the identity element.

The situation is described by the following commutative diagram:

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

We can write the statement of the fundamental theorem on homomorphisms of groups as "Every homomorphic image of a group is isomorphic to a quotient group".

Other versions

Similar theorems are valid for monoids, vector spaces, modules, and rings.

See also

References
  • Beachy, John A. (1999), "Theorem 1.2.7 (The fundamental homomorphism theorem)", Introductory Lectures on Rings and Modules, London Mathematical Society Student Texts, 47, Cambridge University Press, p. 27, ISBN 9780521644075.
  • Grove, Larry C. (2012), "Theorem 1.11 (The Fundamental Homomorphism Theorem)", Algebra, Dover Books on Mathematics, Courier Corporation, p. 11, ISBN 9780486142135.
  • Jacobson, Nathan (2012), "Fundamental theorem on homomorphisms of Ω-algebras", Basic Algebra II, Dover Books on Mathematics (2nd ed.), Courier Corporation, p. 62, ISBN 9780486135212.
  • Rose, John S. (1994), "3.24 Fundamental theorem on homomorphisms", A course on Group Theory [reprint of the 1978 original], Dover Publications, Inc., New York, pp. 44–45, ISBN 0-486-68194-7, MR 1298629.
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