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:

1. 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.

2. 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.