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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 group (mathematics), groups, ring (mathematics), rings, field (mathematics), fields, module (mathe ...
, 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
In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same" ...
is given, and of the kernel
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and image of the homomorphism.
The homomorphism theorem is used to prove the isomorphism theorems
In mathematics, specifically abstract algebra, the isomorphism theorems (also known as Noether's isomorphism theorems) are theorems that describe the relationship between quotients, homomorphisms, and subobjects. Versions of the theorems exist fo ...
.
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
In mathematics, given two groups, (''G'', ∗) and (''H'', ·), a group homomorphism from (''G'', ∗) to (''H'', ·) is a function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that
: h(u*v) = h(u) \cdot h(v)
w ...
, 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
Universal is the adjective for universe.
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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 su ...
.
The situation is described by the following commutative diagram:
:
''h'' is injective if and only if . Therefore, by setting we immediately get the first isomorphism theorem
In mathematics, specifically abstract algebra, the isomorphism theorems (also known as Noether's isomorphism theorems) are theorems that describe the relationship between quotients, homomorphisms, and subobjects. Versions of the theorems exist fo ...
.
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.
Monoid ...
s, vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
s, modules
Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a s ...
, and rings
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.
See also
*Quotient category
In mathematics, a quotient category is a category obtained from another one by identifying sets of morphisms. Formally, it is a quotient object in the category of (locally small) categories, analogous to a quotient group or quotient space, bu ...
References
*. *. *. *{{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 Theoryeprint of the 1978 original
In academic publishing, an eprint or e-print is a digital version of a research document (usually a journal article, but could also be a thesis, conference paper, book chapter, or a book) that is accessible online, usually as green open access, w ...
, url = https://books.google.com/books?id=TWDCAgAAQBAJ&pg=PA44
, year = 1994.
Theorems in abstract algebra