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Given two
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ' ...
, 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
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and image
An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...
of the homomorphism.
The homomorphism theorem is used to prove
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the isomorphism theorems.
Group theoretic version
Given two group
A group is a number of persons or things that are located, gathered, or classed together.
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* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
s ''G'' and ''H'' and a group homomorphism , let ''N'' be a normal subgroup in ''G'' and φ the natural surjective homomorphism (where ''G''/''N'' is the quotient group of ''G'' by ''N''). If ''N'' is a subset of ker(''f'') then there exists a unique homomorphism such that .
In other words, the natural projection φ is universal
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among homomorphisms on ''G'' that map ''N'' to the identity element
In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures ...
.
The situation is described by the following commutative diagram
350px, The commutative diagram used in the proof of the five lemma.
In mathematics, and especially in category theory, a commutative diagram is a diagram such that all directed paths in the diagram with the same start and endpoints lead to the ...
:
:
''h'' is injective if and only if . Therefore, by setting 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 formonoid
In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0.
Monoids a ...
s, vector spaces, modules, and rings.
See also
* Quotient categoryReferences
*. *. *. *{{citation , last = Rose , first = John S. , contribution = 3.24 Fundamental theorem on homomorphisms , isbn = 0-486-68194-7 , mr = 1298629 , pages = 44–45 , publisher = Dover Publications, Inc., New York , title = A course on Group Theory eprint of the 1978 original , url = https://books.google.com/books?id=TWDCAgAAQBAJ&pg=PA44 , year = 1994. Theorems in abstract algebra