Problem 6 - A Book of Abstract Algebra

Question

Let \(G, H\), and \(K\) be groups. Prove the following: For any group \(G\), the function \(f: G \rightarrow G\) defined by \(f(x)=e\) is a homomorphism.

Step-by-Step Solution

Answer

The function \( f(x) = e \) is a homomorphism because it satisfies \( f(ab) = f(a)f(b) \) for all \( a, b \in G \).

Step 1: Define a Homomorphism

A function between two groups, say \( f: G \rightarrow H \), is called a homomorphism if for any elements \( a, b \in G \), the equation \( f(ab) = f(a)f(b) \) holds. We need to check if this holds for the given function \( f(x) = e \), where \( e \) is the identity element of the group \( G \).

Step 2: Apply the Function

Let \( a, b \in G \). We apply the function \( f \) to their product: \( f(ab) = e \), since the function \( f(x) = e \) maps every element to the identity element \( e \) of \( G \).

Step 3: Evaluate Function of Individual Elements

Apply the function to each of the individual elements: \( f(a) = e \) and \( f(b) = e \). Given the nature of the function, this outputs the identity element for both \( a \) and \( b \).

Step 4: Verify Homomorphism Property

Check if \( f(ab) = f(a)f(b) \) holds. We have \( f(ab) = e \) and \( f(a)f(b) = e \cdot e = e \). Since both sides equal \( e \), the homomorphism property holds.

Step 5: Conclude the Proof

Since \( f(ab) = f(a)f(b) \) for all \( a, b \in G \), the function \( f \) is indeed a homomorphism, as it satisfies the condition of the homomorphism property. Conclude that \( f: G \rightarrow G \) defined by \( f(x) = e \) is a homomorphism.

Key Concepts

Identity Element

In group theory, the identity element is a fundamental concept. Every group has an identity element that acts as a neutral element in operations. For a group \( G \), the identity element \( e \) satisfies \( e \cdot a = a \cdot e = a \) for every element \( a \) in the group. The identity element is unique within a group, meaning there is only one such element that fulfills this condition.
For the function \( f: G \rightarrow G \) defined by \( f(x) = e \), the identity element plays a crucial role. The function maps every element \( x \) in \( G \) to the identity element \( e \). This property is what helps establish the homomorphism because the result of any group operation through this function will always be \( e \), maintaining the structure needed for a homomorphism.

• Identity element is unique in any group.
• Acts like the '0' in addition or '1' in multiplication.

Group Theory

Group theory is the study of algebraic structures known as groups. A group consists of a set of elements equipped with an operation that combines any two elements to form a third element. This operation must satisfy four main properties: closure, associativity, identity, and invertibility.
- **Closure** ensures that performing the operation on any two elements of the group produces another element within the group.
- **Associativity** means the way in which three elements are grouped in an operation does not change the result: \( (a \cdot b) \cdot c = a \cdot (b \cdot c) \).
- **Identity** refers to the presence of an element that leaves others unchanged when operated with them.
- **Invertibility** ensures that for each element in the group, there exists another element in the group which combines to form the identity.
This framework is crucial for understanding how functions such as homomorphisms behave, as they must respect and preserve these group operations and properties.

Function Between Groups

A function between groups, especially a homomorphism, connects two groups while preserving their algebraic structure. Such a function, \( f: G \rightarrow H \), needs to satisfy the condition \( f(ab) = f(a)f(b) \) for all \( a, b \in G \). This requirement means that the results of operations in the group \( G \) are mirrored in group \( H \) through the function.
In the exercise, we are given a function \( f: G \rightarrow G \) mapping all elements to the identity \( f(x) = e \). Despite seeming trivial, it's a valid homomorphism. Each element in \( G \) maps to \( e \), keeping the operation results consistent with being acting over the identity.

• Ensures operations in one group reflect correctly in another.
• Maintains the group properties between domain and codomain.
• Preserves structure even if it maps to a single value, the identity element.