# Proof of Homomorphism: f(eG) = eH

• lockedup
In summary, the proof for f(eG) = eH uses the fact that multiplying any element of H by the identity element of H results in that same element.
lockedup

## Homework Statement

If f:G$$\rightarrow$$H is a homomorphism, then f(eG) = eH.

## Homework Equations

The proof from my professor's notes:
f(eG) = f(eG*GeG) = f(eG)*f(eG)
f(eG) = f(eG)*eH
f(eG)*f(eG) = f(eG)*eH
f(eG) = eH

## The Attempt at a Solution

My question is, how do you get from the first line to the second. Because it looks like she's using the proposition to prove itself.

The second line doesn't follow from the first. It simply says that if you multiply any element of H by the identity element of H, you get ... (fill in the rest).

Petek

Petek said:
The second line doesn't follow from the first. It simply says that if you multiply any element of H by the identity element of H, you get ... (fill in the rest).

Petek
I feel like an idiot. Thank you!

## 1. What is "Proof of Homomorphism"?

Proof of Homomorphism is a mathematical concept that describes the relationship between two algebraic structures, such as groups or rings. It states that a function between two structures preserves the algebraic operations of the structures. In simpler terms, it means that the output of the function will follow the same rules as the input.

## 2. What does "f(eG) = eH" mean in "Proof of Homomorphism"?

In "Proof of Homomorphism", f(eG) = eH is known as the identity property. It means that the function f maps the identity element of the first structure (eG) to the identity element of the second structure (eH). This is an important property in proving that a function is a homomorphism.

## 3. How is "Proof of Homomorphism" used in science?

"Proof of Homomorphism" is used in various fields of science, such as computer science, physics, and chemistry. In computer science, homomorphisms are used in cryptography to encrypt and decrypt data. In physics, homomorphisms are used to describe symmetries in physical systems. In chemistry, homomorphisms are used to study the structure and properties of molecules.

## 4. What is the significance of "Proof of Homomorphism" in mathematics?

Homomorphisms are important in mathematics because they help establish connections between different algebraic structures. They also allow for the study of complex structures by breaking them down into simpler, more understandable parts. Additionally, homomorphisms have numerous applications in various branches of mathematics, such as abstract algebra, topology, and number theory.

## 5. What are some real-life examples of "Proof of Homomorphism"?

One real-life example of "Proof of Homomorphism" is the multiplication of complex numbers. When multiplying two complex numbers, the product follows the same rules as the individual numbers, making it a homomorphism. Another example is the rotation of a cube, which is a homomorphism as it preserves the symmetry of the original cube. Homomorphisms can also be seen in the binary operations of addition and multiplication in the integers, where the output follows the same rules as the input.

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