Fundamental theorem on homomorphisms

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In abstract algebra, the fundamental theorem on homomorphisms, also known as the fundamental homomorphism theorem, or the first isomorphism 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 the quotient group of G by K). 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:

Fundamental Homomorphism Theorem.svg

h is injective if and only if K = ker(f). Therefore, 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, vol. 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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