Fundamental theorem on homomorphisms
{{short description|Theorem relating a group with the image and kernel of a homomorphism}}
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. Similar theorems are valid for vector spaces, modules, and rings.
Group-theoretic version
File:Diagram of the fundamental theorem on homomorphisms.svg
Given two groups and and a group homomorphism , let be a normal subgroup in and the natural surjective homomorphism (where is the quotient group of by ). If is a subset of (where represents a kernel) then there exists a unique homomorphism such that .
In other words, the natural projection is universal among homomorphisms on that map to the identity element.
The situation is described by the following commutative diagram:
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".
Proof
The proof follows from two basic facts about homomorphisms, namely their preservation of the group operation, and their mapping of the identity element to the identity element. We need to show that if is a homomorphism of groups, then:
- is a subgroup of {{tmath|1= H }}.
- is isomorphic to {{tmath|1= \text{im}(\phi) }}.
= Proof of 1 =
The operation that is preserved by is the group operation. If {{tmath|1= a, b \in \text{im}(\phi)}}, then there exist elements such that and {{tmath|1= \phi(b')=b}}. For these and {{tmath|1= b }}, we have (since preserves the group operation), and thus, the closure property is satisfied in {{tmath|1= \text{im}(\phi) }}. The identity element is also in because maps the identity element of to it. Since every element in has an inverse such that (because preserves the inverse property as well), we have an inverse for each element in {{tmath|1= \text{im}(\phi) }}, therefore, is a subgroup of {{tmath|1= H }}.
= Proof of 2 =
Construct a map by {{tmath|1= \psi(a\ker(\phi)) = \phi(a) }}. This map is well-defined, as if {{tmath|1= a\ker(\phi) = b\ker(\phi) }}, then and so which gives {{tmath|1= \phi(a) = \phi(b) }}. This map is an isomorphism. is surjective onto by definition. To show injectivity, if , then {{tmath|1= \phi(a) = \phi(b) }}, which implies so {{tmath|1= a\ker(\phi) = b\ker(\phi) }}.
Finally,
:
:
hence preserves the group operation. Hence is an isomorphism between and {{tmath|1= \text{im}(\phi) }}, which completes the proof.
Applications
The group theoretic version of the fundamental homomorphism theorem can be used to show that two selected groups are isomorphic. Two examples are shown below.
= Integers modulo ''n'' =
For each {{tmath|1= n \in \mathbb{N} }}, consider the groups and and a group homomorphism defined by (see modular arithmetic). Next, consider the kernel of {{tmath|1= f }}, {{tmath|1= \text{ker} (f) = n \mathbb{Z} }}, which is a normal subgroup in {{tmath|1= \mathbb{Z} }}. There exists a natural surjective homomorphism defined by {{tmath|1= m \mapsto m+n\mathbb{Z} }}. The theorem asserts that there exists an isomorphism between and {{tmath|1= \mathbb{Z}/n\mathbb{Z} }}, or in other words {{tmath|1= \mathbb{Z}_n \cong \mathbb{Z}/n \mathbb{Z} }}. The commutative diagram is illustrated below.
= ''N / C'' theorem =
Let be a group with subgroup {{tmath|1= H }}. Let {{tmath|1= C_G(H) }}, and be the centralizer, the normalizer and the automorphism group of in {{tmath|1= G }}, respectively. Then, the theorem states that is isomorphic to a subgroup of {{tmath|1= \operatorname{Aut}(H) }}.
== Proof ==
We are able to find a group homomorphism defined by {{tmath|1= g \mapsto ghg^{-1} }}, for all {{tmath|1= h \in H }}. Clearly, the kernel of is {{tmath|1= C_G(H) }}. Hence, we have a natural surjective homomorphism defined by {{tmath|1= g \mapsto gC(H) }}. The fundamental homomorphism theorem then asserts that there exists an isomorphism between and {{tmath|1= \varphi(N_G(H)) }}, which is a subgroup of {{tmath|1= \operatorname{Aut}(H) }}.
See also
References
- {{citation|title=Introductory Lectures on Rings and Modules|volume=47|series=London Mathematical Society Student Texts|first=John A.|last=Beachy|publisher=Cambridge University Press|year=1999|isbn=9780521644075|page=27|url=https://books.google.com/books?id=rnNzivBfgOoC&pg=PA27|contribution=Theorem 1.2.7 (The fundamental homomorphism theorem)}}
- {{citation|title=Algebra|series=Dover Books on Mathematics|first=Larry C.|last=Grove|publisher=Courier Corporation|year=2012|isbn=9780486142135|page=11|url=https://books.google.com/books?id=C4TByeUh9A4C&pg=PA11|contribution=Theorem 1.11 (The Fundamental Homomorphism Theorem)}}
- {{citation|title=Basic Algebra II|edition=2nd|series=Dover Books on Mathematics|first=Nathan|last=Jacobson|publisher=Courier Corporation|year=2012|isbn=9780486135212|url=https://books.google.com/books?id=hn75exNZZ-EC&pg=PA62|page=62|contribution=Fundamental theorem on homomorphisms of Ω-algebras}}
- {{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 [reprint of the 1978 original]
| url = https://books.google.com/books?id=TWDCAgAAQBAJ&pg=PA44
| year = 1994
}}