Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "Homomorphism"

m
(→‎See Also)
 
(One intermediate revision by one other user not shown)
Line 1: Line 1:
 
{{stub}}
 
{{stub}}
  
−
Let <math>A</math> and <math>B</math> be algebraic structures of the same species, for example two [[group]]s or [[field]]s.  A '''homomorphism''' is a [[mapping]] <math>\phi : A \to B</math> that preserves the structure of the species.
+
Let <math>A</math> and <math>B</math> be algebraic structures of the same species, for example two [[group]]s or [[field]]s.  A '''homomorphism''' is a [[function]] <math>\phi : A \to B</math> that preserves the structure of the species.
  
 
For example, if <math>A</math> is a [[substructure]] ([[subgroup]], [[subfield]], etc.) of <math>B</math>, the ''inclusion map'' <math>i: A \to B</math> such that <math>i(a) = a</math> for all <math>a \in A</math> is a homomorphism.
 
For example, if <math>A</math> is a [[substructure]] ([[subgroup]], [[subfield]], etc.) of <math>B</math>, the ''inclusion map'' <math>i: A \to B</math> such that <math>i(a) = a</math> for all <math>a \in A</math> is a homomorphism.
Line 25: Line 25:
 
* [[Endomorphism]]
 
* [[Endomorphism]]
 
* [[Exact Sequence]]
 
* [[Exact Sequence]]
 +
* [[Diffeomorphism]]
  
 
[[Category:Abstract algebra]]
 
[[Category:Abstract algebra]]

Latest revision as of 14:33, 11 February 2024

This article is a stub. Help us out by expanding it.

Let $A$ and $B$ be algebraic structures of the same species, for example two groups or fields. A homomorphism is a function $\phi : A \to B$ that preserves the structure of the species.

For example, if $A$ is a substructure (subgroup, subfield, etc.) of $B$, the inclusion map $i: A \to B$ such that $i(a) = a$ for all $a \in A$ is a homomorphism.

A homomorphism from a structure to itself is called an endomorphism. A homomorphism that is bijective is called an isomorphism. A bijective endomorphism is called an automorphism.

Examples

If $A$ and $B$ are partially ordered sets, a homomorphism from $A$ to $B$ is a mapping $\phi : A \to B$ such that for all $a, b \in A$, if $a \le b$, then $\phi(a) \le \phi(b)$.

If $A$ and $B$ are groups, with group law of $*$, then a homomorphism $\phi : A \to B$ is a mapping such that for all $a,b \in A$, \[\phi( a*b) = \phi(a)* \phi(b) .\] Similarly, if $A$ and $B$ are fields or rings, a homomorphism from $A$ to $B$ is a mapping $\phi : A \to B$ such that for all $a,b \in A$, \begin{align*} \phi(a+b) &= \phi(a) + \phi(b) \\ \phi(ab) &= \phi(a)\phi(b) . \end{align*} In other words, $\phi$ distributes over addition and multiplication.

See Also

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=Homomorphism&oldid=215243"