Morphism in Abstract Algebra

Homomorphism

an operation is called homomorphic if it follows: $\psi (a\oplus b)=\psi (a)\oplus \psi (b)$

Endomorphism

An endomorphism is a homomorphism from an algebraic structure to itself. In other words, it is a map that preserves the structure’s operations and maps elements within the same structure.

Definition

Let $(G,\cdot )$ be a group. An endomorphism of $G$ is a function $\varphi :G\to G$ such that for all $a,b\in G,\varphi (a\cdot b)=\varphi (a)\cdot \varphi (b)$.

Similarly, for a ring $(R,+,\cdot )$ an endomorphism is a function $\psi :R\to R$ such that for all $a,b\in R,\psi (a+b)=\psi (a)+\psi (b)\text{and}\psi (a\cdot b)=\psi (a)\cdot \psi (b)$.

In the context of vector spaces, an endomorphism is a linear transformation from a vector space to itself.

Examples

1. Identity Map: The identity map ${\text{id}}_G:G\to G$ defined by ${\text{id}}_G(x)=x\forall x\in G$ is an endomorphism.

2. Matrix Multiplication: For the ring of $n\times n$ matrices $M_n(\mathbb{R})$, the map $\varphi :M_n(\mathbb{R})\to M_n(\mathbb{R})$ defined by $\varphi (A)=BA$ for a fixed matrix $B$ is an endomorphism.

Isomorphism

An isomorphism is a bijective homomorphism. It is a map between two algebraic structures that preserves operations and has an inverse that is also a homomorphism.

Definition

Let $(G,\cdot )$ and $(H,\ast )$ be groups. An isomorphism from $G\to H$ is a bijective function $\varphi :G\to H$ such that $\forall a,b\in G,\varphi (a\cdot b)=\varphi (a)\ast \varphi (b)$

Similarly, for rings $(R,+,\cdot )$ and $(S,\oplus ,\otimes )$, an isomorphism is a bijective function $\psi :R\to S$ such that for all $a,b\in R,\psi (a+b)=\psi (a)\oplus \psi (b)\text{and}\psi (a\cdot b)=\psi (a)\otimes \psi (b)$

In the context of vector spaces, an isomorphism is a bijective linear transformation.

Examples

1. Vector Spaces: The map $\varphi :{\mathbb{R}}^2\to {\mathbb{R}}^2$ defined by $\varphi (x,y)=(x+y,x-y)$ is an isomorphism between the vector spaces ${\mathbb{R}}^2$ and itself.

2. Groups: The map $\varphi :\mathbb{Z}\to 2\mathbb{Z}$ defined by $\varphi (n)=2n$ is an isomorphism between the group of integers under addition $\mathbb{Z}$ and the group of even integers under addition $2\mathbb{Z}$.

Properties of Isomorphisms

1. Bijectivity: An isomorphism is both injective (one-to-one) and surjective (onto).
2. Preservation of Structure: Isomorphisms preserve the algebraic operations, meaning that the structure of the original algebraic system is retained in the image.
3. Inverse Map: The inverse of an isomorphism is also an isomorphism. If $\varphi :G\to H$ is an isomorphism, then there exists a map ${\varphi }^{-1}:H\to G$ such that ${\varphi }^{-1}\circ \varphi ={\text{id}}_G$ and $\varphi \circ {\varphi }^{-1}={\text{id}}_H$.

Relation Between Endomorphisms and Isomorphisms

• Endomorphism: A map from a structure to itself that preserves the operations. Not necessarily bijective.
• Isomorphism: A bijective homomorphism between two structures that preserves the operations. An isomorphism between a structure and itself is called an automorphism.

Automorphisms

An automorphism is a special case of an isomorphism where the domain and codomain are the same.

Definition

An automorphism of a group $G$ is an isomorphism from $G$ to itself. Formally, $\varphi :G\to G$ is an automorphism if it is a bijective homomorphism.

Examples

1. Identity Map: The identity map ${\text{id}}_G$ is an automorphism of any group $G$.
2. Complex Conjugation: The map $\varphi :\mathbb{C}\to \mathbb{C}$ defined by $\varphi (z)=\overline{z}$ (complex conjugation) is an automorphism of the field of complex numbers.

Automorphisms form a group under composition, known as the automorphism group of the structure.