# 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

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.

## 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.