Mathematics > Abstract Algebra > Homomorphisms
Description:
Homomorphisms are fundamental concepts in the field of abstract algebra, which itself is a branch of mathematics that studies algebraic structures such as groups, rings, and fields. A homomorphism provides a way to map between two algebraic structures of the same type while preserving the operations that define them.
Definition:
A homomorphism is a function \( \varphi \) between two algebraic structures (such as groups, rings, or vector spaces) that respects the structure’s operations. Specifically, if we consider two groups \( (G, *) \) and \( (H, \cdot) \), a group homomorphism from \( G \) to \( H \) is a function \( \varphi: G \to H \) such that for all elements \( a, b \in G \),
\[ \varphi(a * b) = \varphi(a) \cdot \varphi(b). \]
This equation indicates that the image under \( \varphi \) of the product \( a * b \) in \( G \) is equal to the product of the images \( \varphi(a) \) and \( \varphi(b) \) in \( H \). Essentially, the function \( \varphi \) preserves the group operation.
Properties of Homomorphisms:
Preservation of Identity Element:
Any homomorphism \( \varphi \) maps the identity element of the group \( G \) to the identity element of the group \( H \). That is,\[ \varphi(e_G) = e_H, \]
where \( e_G \) and \( e_H \) are the identity elements in \( G \) and \( H \), respectively.
Compatibility with Inverses:
For each element \( a \in G \), the homomorphism \( \varphi \) maps the inverse of \( a \) in \( G \) to the inverse of \( \varphi(a) \) in \( H \):\[ \varphi(a^{-1}) = \varphi(a)^{-1}. \]
Examples:
Group Homomorphisms:
Consider the groups \( ( \mathbb{Z}, +) \) and \( ( \mathbb{Z}_6, +_6) \). The function \( \varphi: \mathbb{Z} \to \mathbb{Z}_6 \) defined by \( \varphi(n) = n \mod 6 \) is a homomorphism because it preserves the addition operation.Ring Homomorphisms:
For rings \( (R, +, \cdot) \) and \( (S, \oplus, \otimes) \), a ring homomorphism \( \psi: R \to S \) must satisfy both:\[ \psi(a + b) = \psi(a) \oplus \psi(b) \]
\[ \psi(a \cdot b) = \psi(a) \otimes \psi(b). \]Linear Transformations:
Linear transformations in vector spaces are essentially homomorphisms between vector spaces. A function \( T: V \to W \) between vector spaces \( V \) and \( W \) over the same field \( F \) is a linear transformation if for all \( \mathbf{u}, \mathbf{v} \in V \) and \( c \in F \),\[ T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v}) \]
\[ T(c \mathbf{u}) = c T(\mathbf{u}). \]
Homomorphisms are fundamental in connecting and comparing different algebraic structures, making them a cornerstone concept in abstract algebra. Through homomorphisms, one can study properties that are preserved across different structures, thereby drawing deeper insights into the nature of the structures themselves.